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Theorem bitsf1 15215
 Description: The bits function is an injection from ℤ to 𝒫 ℕ0. It is obviously not a bijection (by Cantor's theorem canth2 8154), and in fact its range is the set of finite and cofinite subsets of ℕ0. (Contributed by Mario Carneiro, 22-Sep-2016.)
Assertion
Ref Expression
bitsf1 bits:ℤ–1-1→𝒫 ℕ0

Proof of Theorem bitsf1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bitsf 15196 . 2 bits:ℤ⟶𝒫 ℕ0
2 simpl 472 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
32zcnd 11521 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ)
43adantr 480 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℂ)
5 simpr 476 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
65zcnd 11521 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
76adantr 480 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℂ)
84negcld 10417 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 ∈ ℂ)
97negcld 10417 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℂ)
10 1cnd 10094 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 1 ∈ ℂ)
11 simprr 811 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
1211difeq2d 3761 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (ℕ0 ∖ (bits‘𝑦)))
13 bitscmp 15207 . . . . . . . . . . 11 (𝑥 ∈ ℤ → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
1413ad2antrr 762 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
15 bitscmp 15207 . . . . . . . . . . 11 (𝑦 ∈ ℤ → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
1615ad2antlr 763 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
1712, 14, 163eqtr3d 2693 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) = (bits‘(-𝑦 − 1)))
18 nnm1nn0 11372 . . . . . . . . . . 11 (-𝑥 ∈ ℕ → (-𝑥 − 1) ∈ ℕ0)
1918ad2antrl 764 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) ∈ ℕ0)
20 fvres 6245 . . . . . . . . . 10 ((-𝑥 − 1) ∈ ℕ0 → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
2119, 20syl 17 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
22 ominf 8213 . . . . . . . . . . . . . . . . 17 ¬ ω ∈ Fin
23 nn0ennn 12818 . . . . . . . . . . . . . . . . . . 19 0 ≈ ℕ
24 nnenom 12819 . . . . . . . . . . . . . . . . . . 19 ℕ ≈ ω
2523, 24entr2i 8052 . . . . . . . . . . . . . . . . . 18 ω ≈ ℕ0
26 enfii 8218 . . . . . . . . . . . . . . . . . 18 ((ℕ0 ∈ Fin ∧ ω ≈ ℕ0) → ω ∈ Fin)
2725, 26mpan2 707 . . . . . . . . . . . . . . . . 17 (ℕ0 ∈ Fin → ω ∈ Fin)
2822, 27mto 188 . . . . . . . . . . . . . . . 16 ¬ ℕ0 ∈ Fin
29 difinf 8271 . . . . . . . . . . . . . . . 16 ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑥) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
3028, 29mpan 706 . . . . . . . . . . . . . . 15 ((bits‘𝑥) ∈ Fin → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
31 bitsfi 15206 . . . . . . . . . . . . . . . . 17 ((-𝑥 − 1) ∈ ℕ0 → (bits‘(-𝑥 − 1)) ∈ Fin)
3219, 31syl 17 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) ∈ Fin)
3314, 32eqeltrd 2730 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
3430, 33nsyl3 133 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑥) ∈ Fin)
3511, 34eqneltrrd 2750 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑦) ∈ Fin)
36 bitsfi 15206 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → (bits‘𝑦) ∈ Fin)
3735, 36nsyl 135 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ 𝑦 ∈ ℕ0)
385znegcld 11522 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
39 elznn 11431 . . . . . . . . . . . . . . . . 17 (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℝ ∧ (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0)))
4039simprbi 479 . . . . . . . . . . . . . . . 16 (-𝑦 ∈ ℤ → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
4138, 40syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
426negnegd 10421 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑦 = 𝑦)
4342eleq1d 2715 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑦 ∈ ℕ0𝑦 ∈ ℕ0))
4443orbi2d 738 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0) ↔ (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0)))
4541, 44mpbid 222 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
4645adantr 480 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
4746ord 391 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
4837, 47mt3d 140 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℕ)
49 nnm1nn0 11372 . . . . . . . . . . 11 (-𝑦 ∈ ℕ → (-𝑦 − 1) ∈ ℕ0)
5048, 49syl 17 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 − 1) ∈ ℕ0)
51 fvres 6245 . . . . . . . . . 10 ((-𝑦 − 1) ∈ ℕ0 → ((bits ↾ ℕ0)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
5250, 51syl 17 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
5317, 21, 523eqtr4d 2695 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)))
54 bitsf1o 15214 . . . . . . . . . . 11 (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)
55 f1of1 6174 . . . . . . . . . . 11 ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) → (bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin))
5654, 55ax-mp 5 . . . . . . . . . 10 (bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin)
57 f1fveq 6559 . . . . . . . . . 10 (((bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin) ∧ ((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0)) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
5856, 57mpan 706 . . . . . . . . 9 (((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
5919, 50, 58syl2anc 694 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
6053, 59mpbid 222 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) = (-𝑦 − 1))
618, 9, 10, 60subcan2d 10472 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 = -𝑦)
624, 7, 61neg11d 10442 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
6362expr 642 . . . 4 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ -𝑥 ∈ ℕ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
643negnegd 10421 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑥 = 𝑥)
6564eleq1d 2715 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑥 ∈ ℕ0𝑥 ∈ ℕ0))
6665biimpa 500 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
67 simprr 811 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
68 fvres 6245 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
6968ad2antrl 764 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
7015ad2antlr 763 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
71 bitsfi 15206 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℕ0 → (bits‘𝑥) ∈ Fin)
7271ad2antrl 764 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) ∈ Fin)
7367, 72eqeltrrd 2731 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑦) ∈ Fin)
74 difinf 8271 . . . . . . . . . . . . . 14 ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑦) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
7528, 73, 74sylancr 696 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
7670, 75eqneltrrd 2750 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘(-𝑦 − 1)) ∈ Fin)
77 bitsfi 15206 . . . . . . . . . . . 12 ((-𝑦 − 1) ∈ ℕ0 → (bits‘(-𝑦 − 1)) ∈ Fin)
7876, 77nsyl 135 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (-𝑦 − 1) ∈ ℕ0)
7978, 49nsyl 135 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ -𝑦 ∈ ℕ)
8045adantr 480 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
8180ord 391 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
8279, 81mpd 15 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℕ0)
83 fvres 6245 . . . . . . . . 9 (𝑦 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑦) = (bits‘𝑦))
8482, 83syl 17 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑦) = (bits‘𝑦))
8567, 69, 843eqtr4d 2695 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦))
86 simprl 809 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℕ0)
87 f1fveq 6559 . . . . . . . . 9 (((bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin) ∧ (𝑥 ∈ ℕ0𝑦 ∈ ℕ0)) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8856, 87mpan 706 . . . . . . . 8 ((𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8986, 82, 88syl2anc 694 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
9085, 89mpbid 222 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
9190expr 642 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
9266, 91syldan 486 . . . 4 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
932znegcld 11522 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑥 ∈ ℤ)
94 elznn 11431 . . . . . 6 (-𝑥 ∈ ℤ ↔ (-𝑥 ∈ ℝ ∧ (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0)))
9594simprbi 479 . . . . 5 (-𝑥 ∈ ℤ → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
9693, 95syl 17 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
9763, 92, 96mpjaodan 844 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
9897rgen2a 3006 . 2 𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)
99 dff13 6552 . 2 (bits:ℤ–1-1→𝒫 ℕ0 ↔ (bits:ℤ⟶𝒫 ℕ0 ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)))
1001, 98, 99mpbir2an 975 1 bits:ℤ–1-1→𝒫 ℕ0
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ∖ cdif 3604   ∩ cin 3606  𝒫 cpw 4191   class class class wbr 4685   ↾ cres 5145  ⟶wf 5922  –1-1→wf1 5923  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690  ωcom 7107   ≈ cen 7994  Fincfn 7997  ℂcc 9972  ℝcr 9973  1c1 9975   − cmin 10304  -cneg 10305  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  bitscbits 15188 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-dvds 15028  df-bits 15191 This theorem is referenced by:  bitsuz  15243  eulerpartlemmf  30565
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