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Theorem bitsf1 16326
Description: The bits function is an injection from to 𝒫 ℕ0. It is obviously not a bijection (by Cantor's theorem canth2 9074), 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 16307 . 2 bits:ℤ⟶𝒫 ℕ0
2 simpl 483 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
32zcnd 12608 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ)
43adantr 481 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℂ)
5 simpr 485 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
65zcnd 12608 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
76adantr 481 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℂ)
84negcld 11499 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 ∈ ℂ)
97negcld 11499 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℂ)
10 1cnd 11150 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 1 ∈ ℂ)
11 simprr 771 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
1211difeq2d 4082 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (ℕ0 ∖ (bits‘𝑦)))
13 bitscmp 16318 . . . . . . . . . . 11 (𝑥 ∈ ℤ → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
1413ad2antrr 724 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
15 bitscmp 16318 . . . . . . . . . . 11 (𝑦 ∈ ℤ → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
1615ad2antlr 725 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
1712, 14, 163eqtr3d 2784 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) = (bits‘(-𝑦 − 1)))
18 nnm1nn0 12454 . . . . . . . . . . 11 (-𝑥 ∈ ℕ → (-𝑥 − 1) ∈ ℕ0)
1918ad2antrl 726 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) ∈ ℕ0)
2019fvresd 6862 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
21 ominf 9202 . . . . . . . . . . . . . . . . 17 ¬ ω ∈ Fin
22 nn0ennn 13884 . . . . . . . . . . . . . . . . . . 19 0 ≈ ℕ
23 nnenom 13885 . . . . . . . . . . . . . . . . . . 19 ℕ ≈ ω
2422, 23entr2i 8949 . . . . . . . . . . . . . . . . . 18 ω ≈ ℕ0
25 enfii 9133 . . . . . . . . . . . . . . . . . 18 ((ℕ0 ∈ Fin ∧ ω ≈ ℕ0) → ω ∈ Fin)
2624, 25mpan2 689 . . . . . . . . . . . . . . . . 17 (ℕ0 ∈ Fin → ω ∈ Fin)
2721, 26mto 196 . . . . . . . . . . . . . . . 16 ¬ ℕ0 ∈ Fin
28 difinf 9260 . . . . . . . . . . . . . . . 16 ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑥) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
2927, 28mpan 688 . . . . . . . . . . . . . . 15 ((bits‘𝑥) ∈ Fin → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
30 bitsfi 16317 . . . . . . . . . . . . . . . . 17 ((-𝑥 − 1) ∈ ℕ0 → (bits‘(-𝑥 − 1)) ∈ Fin)
3119, 30syl 17 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) ∈ Fin)
3214, 31eqeltrd 2838 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
3329, 32nsyl3 138 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑥) ∈ Fin)
3411, 33eqneltrrd 2858 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑦) ∈ Fin)
35 bitsfi 16317 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → (bits‘𝑦) ∈ Fin)
3634, 35nsyl 140 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ 𝑦 ∈ ℕ0)
375znegcld 12609 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
38 elznn 12515 . . . . . . . . . . . . . . . . 17 (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℝ ∧ (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0)))
3938simprbi 497 . . . . . . . . . . . . . . . 16 (-𝑦 ∈ ℤ → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
4037, 39syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
416negnegd 11503 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑦 = 𝑦)
4241eleq1d 2822 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑦 ∈ ℕ0𝑦 ∈ ℕ0))
4342orbi2d 914 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0) ↔ (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0)))
4440, 43mpbid 231 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
4544adantr 481 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
4645ord 862 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
4736, 46mt3d 148 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℕ)
48 nnm1nn0 12454 . . . . . . . . . . 11 (-𝑦 ∈ ℕ → (-𝑦 − 1) ∈ ℕ0)
4947, 48syl 17 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 − 1) ∈ ℕ0)
5049fvresd 6862 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
5117, 20, 503eqtr4d 2786 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)))
52 bitsf1o 16325 . . . . . . . . . . 11 (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)
53 f1of1 6783 . . . . . . . . . . 11 ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) → (bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin))
5452, 53ax-mp 5 . . . . . . . . . 10 (bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin)
55 f1fveq 7209 . . . . . . . . . 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)))
5654, 55mpan 688 . . . . . . . . 9 (((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
5719, 49, 56syl2anc 584 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
5851, 57mpbid 231 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) = (-𝑦 − 1))
598, 9, 10, 58subcan2d 11554 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 = -𝑦)
604, 7, 59neg11d 11524 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
6160expr 457 . . . 4 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ -𝑥 ∈ ℕ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
623negnegd 11503 . . . . . . 7 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑥 = 𝑥)
6362eleq1d 2822 . . . . . 6 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑥 ∈ ℕ0𝑥 ∈ ℕ0))
6463biimpa 477 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
65 simprr 771 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
66 fvres 6861 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
6766ad2antrl 726 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
6815ad2antlr 725 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
69 bitsfi 16317 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℕ0 → (bits‘𝑥) ∈ Fin)
7069ad2antrl 726 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) ∈ Fin)
7165, 70eqeltrrd 2839 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑦) ∈ Fin)
72 difinf 9260 . . . . . . . . . . . . . 14 ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑦) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
7327, 71, 72sylancr 587 . . . . . . . . . . . . 13 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
7468, 73eqneltrrd 2858 . . . . . . . . . . . 12 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘(-𝑦 − 1)) ∈ Fin)
75 bitsfi 16317 . . . . . . . . . . . 12 ((-𝑦 − 1) ∈ ℕ0 → (bits‘(-𝑦 − 1)) ∈ Fin)
7674, 75nsyl 140 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (-𝑦 − 1) ∈ ℕ0)
7776, 48nsyl 140 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ -𝑦 ∈ ℕ)
7844adantr 481 . . . . . . . . . . 11 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
7978ord 862 . . . . . . . . . 10 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
8077, 79mpd 15 . . . . . . . . 9 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℕ0)
8180fvresd 6862 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑦) = (bits‘𝑦))
8265, 67, 813eqtr4d 2786 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦))
83 simprl 769 . . . . . . . 8 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℕ0)
84 f1fveq 7209 . . . . . . . . 9 (((bits ↾ ℕ0):ℕ01-1→(𝒫 ℕ0 ∩ Fin) ∧ (𝑥 ∈ ℕ0𝑦 ∈ ℕ0)) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8554, 84mpan 688 . . . . . . . 8 ((𝑥 ∈ ℕ0𝑦 ∈ ℕ0) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8683, 80, 85syl2anc 584 . . . . . . 7 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
8782, 86mpbid 231 . . . . . 6 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
8887expr 457 . . . . 5 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
8964, 88syldan 591 . . . 4 (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
902znegcld 12609 . . . . 5 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑥 ∈ ℤ)
91 elznn 12515 . . . . . 6 (-𝑥 ∈ ℤ ↔ (-𝑥 ∈ ℝ ∧ (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0)))
9291simprbi 497 . . . . 5 (-𝑥 ∈ ℤ → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
9390, 92syl 17 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
9461, 89, 93mpjaodan 957 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
9594rgen2 3194 . 2 𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)
96 dff13 7202 . 2 (bits:ℤ–1-1→𝒫 ℕ0 ↔ (bits:ℤ⟶𝒫 ℕ0 ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦)))
971, 95, 96mpbir2an 709 1 bits:ℤ–1-1→𝒫 ℕ0
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3064  cdif 3907  cin 3909  𝒫 cpw 4560   class class class wbr 5105  cres 5635  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  ωcom 7802  cen 8880  Fincfn 8883  cc 11049  cr 11050  1c1 11052  cmin 11385  -cneg 11386  cn 12153  0cn0 12413  cz 12499  bitscbits 16299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-bits 16302
This theorem is referenced by:  bitsuz  16354  eulerpartlemmf  32975
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