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Theorem bitsf1 15541
Description: The bits function is an injection from to 𝒫 ℕ0. It is obviously not a bijection (by Cantor's theorem canth2 8382), and in fact its range is the set of finite and cofinite subsets of 0. (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 15522 . 2 bits:ℤ⟶𝒫 ℕ0
2 simpl 476 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
32zcnd 11811 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ)
43adantr 474 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℂ)
5 simpr 479 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
65zcnd 11811 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
76adantr 474 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℂ)
84negcld 10700 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 ∈ ℂ)
97negcld 10700 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℂ)
10 1cnd 10351 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 1 ∈ ℂ)
11 simprr 791 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
1211difeq2d 3955 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (ℕ0 ∖ (bits‘𝑦)))
13 bitscmp 15533 . . . . . . . . . . 11 (𝑥 ∈ ℤ → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
1413ad2antrr 719 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
15 bitscmp 15533 . . . . . . . . . . 11 (𝑦 ∈ ℤ → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
1615ad2antlr 720 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
1712, 14, 163eqtr3d 2869 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) = (bits‘(-𝑦 − 1)))
18 nnm1nn0 11661 . . . . . . . . . . 11 (-𝑥 ∈ ℕ → (-𝑥 − 1) ∈ ℕ0)
1918ad2antrl 721 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) ∈ ℕ0)
20 fvres 6452 . . . . . . . . . 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 8441 . . . . . . . . . . . . . . . . 17 ¬ ω ∈ Fin
23 nn0ennn 13073 . . . . . . . . . . . . . . . . . . 19 0 ≈ ℕ
24 nnenom 13074 . . . . . . . . . . . . . . . . . . 19 ℕ ≈ ω
2523, 24entr2i 8277 . . . . . . . . . . . . . . . . . 18 ω ≈ ℕ0
26 enfii 8446 . . . . . . . . . . . . . . . . . 18 ((ℕ0 ∈ Fin ∧ ω ≈ ℕ0) → ω ∈ Fin)
2725, 26mpan2 684 . . . . . . . . . . . . . . . . 17 (ℕ0 ∈ Fin → ω ∈ Fin)
2822, 27mto 189 . . . . . . . . . . . . . . . 16 ¬ ℕ0 ∈ Fin
29 difinf 8499 . . . . . . . . . . . . . . . 16 ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑥) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
3028, 29mpan 683 . . . . . . . . . . . . . . 15 ((bits‘𝑥) ∈ Fin → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
31 bitsfi 15532 . . . . . . . . . . . . . . . . 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 2906 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
3430, 33nsyl3 136 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑥) ∈ Fin)
3511, 34eqneltrrd 2926 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑦) ∈ Fin)
36 bitsfi 15532 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → (bits‘𝑦) ∈ Fin)
3735, 36nsyl 138 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ 𝑦 ∈ ℕ0)
385znegcld 11812 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
39 elznn 11720 . . . . . . . . . . . . . . . . 17 (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℝ ∧ (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0)))
4039simprbi 492 . . . . . . . . . . . . . . . 16 (-𝑦 ∈ ℤ → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
4138, 40syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
426negnegd 10704 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑦 = 𝑦)
4342eleq1d 2891 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑦 ∈ ℕ0𝑦 ∈ ℕ0))
4443orbi2d 946 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0) ↔ (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0)))
4541, 44mpbid 224 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
4645adantr 474 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
4746ord 897 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
4837, 47mt3d 143 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℕ)
49 nnm1nn0 11661 . . . . . . . . . . 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 6452 . . . . . . . . . 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 2871 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)))
54 bitsf1o 15540 . . . . . . . . . . 11 (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)
55 f1of1 6377 . . . . . . . . . . 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 6774 . . . . . . . . . 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 683 . . . . . . . . 9 (((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
5919, 50, 58syl2anc 581 . . . . . . . 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 224 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) = (-𝑦 − 1))
618, 9, 10, 60subcan2d 10755 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 = -𝑦)
624, 7, 61neg11d 10725 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
6362expr 450 . . . 4 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ -𝑥 ∈ ℕ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
643negnegd 10704 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑥 = 𝑥)
6564eleq1d 2891 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑥 ∈ ℕ0𝑥 ∈ ℕ0))
6665biimpa 470 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
67 simprr 791 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
68 fvres 6452 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
6968ad2antrl 721 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
7015ad2antlr 720 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
71 bitsfi 15532 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℕ0 → (bits‘𝑥) ∈ Fin)
7271ad2antrl 721 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) ∈ Fin)
7367, 72eqeltrrd 2907 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑦) ∈ Fin)
74 difinf 8499 . . . . . . . . . . . . . 14 ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑦) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
7528, 73, 74sylancr 583 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
7670, 75eqneltrrd 2926 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘(-𝑦 − 1)) ∈ Fin)
77 bitsfi 15532 . . . . . . . . . . . 12 ((-𝑦 − 1) ∈ ℕ0 → (bits‘(-𝑦 − 1)) ∈ Fin)
7876, 77nsyl 138 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (-𝑦 − 1) ∈ ℕ0)
7978, 49nsyl 138 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ -𝑦 ∈ ℕ)
8045adantr 474 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
8180ord 897 . . . . . . . . . 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 6452 . . . . . . . . 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 2871 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦))
86 simprl 789 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℕ0)
87 f1fveq 6774 . . . . . . . . 9 (((bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin) ∧ (𝑥 ∈ ℕ0𝑦 ∈ ℕ0)) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8856, 87mpan 683 . . . . . . . 8 ((𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8986, 82, 88syl2anc 581 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
9085, 89mpbid 224 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
9190expr 450 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
9266, 91syldan 587 . . . 4 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
932znegcld 11812 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑥 ∈ ℤ)
94 elznn 11720 . . . . . 6 (-𝑥 ∈ ℤ ↔ (-𝑥 ∈ ℝ ∧ (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0)))
9594simprbi 492 . . . . 5 (-𝑥 ∈ ℤ → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
9693, 95syl 17 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
9763, 92, 96mpjaodan 988 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
9897rgen2a 3186 . 2 𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)
99 dff13 6767 . 2 (bits:ℤ–1-1→𝒫 ℕ0 ↔ (bits:ℤ⟶𝒫 ℕ0 ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)))
1001, 98, 99mpbir2an 704 1 bits:ℤ–1-1→𝒫 ℕ0
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 880   = wceq 1658  wcel 2166  wral 3117  cdif 3795  cin 3797  𝒫 cpw 4378   class class class wbr 4873  cres 5344  wf 6119  1-1wf1 6120  1-1-ontowf1o 6122  cfv 6123  (class class class)co 6905  ωcom 7326  cen 8219  Fincfn 8222  cc 10250  cr 10251  1c1 10253  cmin 10585  -cneg 10586  cn 11350  0cn0 11618  cz 11704  bitscbits 15514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-disj 4842  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-n0 11619  df-xnn0 11691  df-z 11705  df-uz 11969  df-rp 12113  df-fz 12620  df-fzo 12761  df-fl 12888  df-mod 12964  df-seq 13096  df-exp 13155  df-hash 13411  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-clim 14596  df-sum 14794  df-dvds 15358  df-bits 15517
This theorem is referenced by:  bitsuz  15569  eulerpartlemmf  30982
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