Theorem bitsf1 16406

Description: The bits function is an injection from ℤ to 𝒫 ℕ₀. It is obviously not a bijection (by Cantor's theorem canth2 9061), 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.

Step	Hyp	Ref	Expression
1		bitsf 16387	. 2 ⊢ bits:ℤ⟶𝒫 ℕ0
2		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
3	2	zcnd 12625	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℂ)
4	3	adantr 480	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℂ)
5		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
6	5	zcnd 12625	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℂ)
7	6	adantr 480	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℂ)
8	4	negcld 11483	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 ∈ ℂ)
9	7	negcld 11483	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℂ)
10		1cnd 11130	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 1 ∈ ℂ)
11		simprr 773	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
12	11	difeq2d 4067	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (ℕ0 ∖ (bits‘𝑦)))
13		bitscmp 16398	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
14	13	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) = (bits‘(-𝑥 − 1)))
15		bitscmp 16398	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℤ → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
16	15	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
17	12, 14, 16	3eqtr3d 2780	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) = (bits‘(-𝑦 − 1)))
18		nnm1nn0 12469	. . . . . . . . . . 11 ⊢ (-𝑥 ∈ ℕ → (-𝑥 − 1) ∈ ℕ0)
19	18	ad2antrl 729	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) ∈ ℕ0)
20	19	fvresd 6854	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = (bits‘(-𝑥 − 1)))
21		ominf 9167	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ω ∈ Fin
22		nn0ennn 13932	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 ≈ ℕ
23		nnenom 13933	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ ≈ ω
24	22, 23	entr2i 8949	. . . . . . . . . . . . . . . . . 18 ⊢ ω ≈ ℕ0
25		enfii 9113	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℕ0 ∈ Fin ∧ ω ≈ ℕ0) → ω ∈ Fin)
26	24, 25	mpan2 692	. . . . . . . . . . . . . . . . 17 ⊢ (ℕ0 ∈ Fin → ω ∈ Fin)
27	21, 26	mto 197	. . . . . . . . . . . . . . . 16 ⊢ ¬ ℕ0 ∈ Fin
28		difinf 9214	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑥) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
29	27, 28	mpan 691	. . . . . . . . . . . . . . 15 ⊢ ((bits‘𝑥) ∈ Fin → ¬ (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
30		bitsfi 16397	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑥 − 1) ∈ ℕ0 → (bits‘(-𝑥 − 1)) ∈ Fin)
31	19, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘(-𝑥 − 1)) ∈ Fin)
32	14, 31	eqeltrd 2837	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑥)) ∈ Fin)
33	29, 32	nsyl3 138	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑥) ∈ Fin)
34	11, 33	eqneltrrd 2858	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘𝑦) ∈ Fin)
35		bitsfi 16397	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ0 → (bits‘𝑦) ∈ Fin)
36	34, 35	nsyl 140	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ 𝑦 ∈ ℕ0)
37	5	znegcld 12626	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ)
38		elznn 12531	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℝ ∧ (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0)))
39	38	simprbi 497	. . . . . . . . . . . . . . . 16 ⊢ (-𝑦 ∈ ℤ → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
40	37, 39	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0))
41	6	negnegd 11487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑦 = 𝑦)
42	41	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑦 ∈ ℕ0 ↔ 𝑦 ∈ ℕ0))
43	42	orbi2d 916	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((-𝑦 ∈ ℕ ∨ --𝑦 ∈ ℕ0) ↔ (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0)))
44	40, 43	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
46	45	ord 865	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
47	36, 46	mt3d 148	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑦 ∈ ℕ)
48		nnm1nn0 12469	. . . . . . . . . . 11 ⊢ (-𝑦 ∈ ℕ → (-𝑦 − 1) ∈ ℕ0)
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 − 1) ∈ ℕ0)
50	49	fvresd 6854	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑦 − 1)) = (bits‘(-𝑦 − 1)))
51	17, 20, 50	3eqtr4d 2782	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)))
52		bitsf1o 16405	. . . . . . . . . . 11 ⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin)
53		f1of1 6773	. . . . . . . . . . 11 ⊢ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) → (bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin))
54	52, 53	ax-mp 5	. . . . . . . . . 10 ⊢ (bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin)
55		f1fveq 7210	. . . . . . . . . 10 ⊢ (((bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin) ∧ ((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0)) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
56	54, 55	mpan 691	. . . . . . . . 9 ⊢ (((-𝑥 − 1) ∈ ℕ0 ∧ (-𝑦 − 1) ∈ ℕ0) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
57	19, 49, 56	syl2anc 585	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘(-𝑥 − 1)) = ((bits ↾ ℕ0)‘(-𝑦 − 1)) ↔ (-𝑥 − 1) = (-𝑦 − 1)))
58	51, 57	mpbid 232	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑥 − 1) = (-𝑦 − 1))
59	8, 9, 10, 58	subcan2d 11538	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → -𝑥 = -𝑦)
60	4, 7, 59	neg11d 11508	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (-𝑥 ∈ ℕ ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
61	60	expr 456	. . . 4 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ -𝑥 ∈ ℕ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
62	3	negnegd 11487	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → --𝑥 = 𝑥)
63	62	eleq1d 2822	. . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (--𝑥 ∈ ℕ0 ↔ 𝑥 ∈ ℕ0))
64	63	biimpa 476	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
65		simprr 773	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) = (bits‘𝑦))
66		fvres 6853	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
67	66	ad2antrl 729	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = (bits‘𝑥))
68	15	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (ℕ0 ∖ (bits‘𝑦)) = (bits‘(-𝑦 − 1)))
69		bitsfi 16397	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (bits‘𝑥) ∈ Fin)
70	69	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑥) ∈ Fin)
71	65, 70	eqeltrrd 2838	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (bits‘𝑦) ∈ Fin)
72		difinf 9214	. . . . . . . . . . . . . 14 ⊢ ((¬ ℕ0 ∈ Fin ∧ (bits‘𝑦) ∈ Fin) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
73	27, 71, 72	sylancr 588	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (ℕ0 ∖ (bits‘𝑦)) ∈ Fin)
74	68, 73	eqneltrrd 2858	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (bits‘(-𝑦 − 1)) ∈ Fin)
75		bitsfi 16397	. . . . . . . . . . . 12 ⊢ ((-𝑦 − 1) ∈ ℕ0 → (bits‘(-𝑦 − 1)) ∈ Fin)
76	74, 75	nsyl 140	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ (-𝑦 − 1) ∈ ℕ0)
77	76, 48	nsyl 140	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ¬ -𝑦 ∈ ℕ)
78	44	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (-𝑦 ∈ ℕ ∨ 𝑦 ∈ ℕ0))
79	78	ord 865	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (¬ -𝑦 ∈ ℕ → 𝑦 ∈ ℕ0))
80	77, 79	mpd 15	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑦 ∈ ℕ0)
81	80	fvresd 6854	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑦) = (bits‘𝑦))
82	65, 67, 81	3eqtr4d 2782	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → ((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦))
83		simprl 771	. . . . . . . 8 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 ∈ ℕ0)
84		f1fveq 7210	. . . . . . . . 9 ⊢ (((bits ↾ ℕ0):ℕ0–1-1→(𝒫 ℕ0 ∩ Fin) ∧ (𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
85	54, 84	mpan 691	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
86	83, 80, 85	syl2anc 585	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → (((bits ↾ ℕ0)‘𝑥) = ((bits ↾ ℕ0)‘𝑦) ↔ 𝑥 = 𝑦))
87	82, 86	mpbid 232	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑥 ∈ ℕ0 ∧ (bits‘𝑥) = (bits‘𝑦))) → 𝑥 = 𝑦)
88	87	expr 456	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
89	64, 88	syldan 592	. . . 4 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ --𝑥 ∈ ℕ0) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
90	2	znegcld 12626	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → -𝑥 ∈ ℤ)
91		elznn 12531	. . . . . 6 ⊢ (-𝑥 ∈ ℤ ↔ (-𝑥 ∈ ℝ ∧ (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0)))
92	91	simprbi 497	. . . . 5 ⊢ (-𝑥 ∈ ℤ → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
93	90, 92	syl 17	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (-𝑥 ∈ ℕ ∨ --𝑥 ∈ ℕ0))
94	61, 89, 93	mpjaodan 961	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((bits‘𝑥) = (bits‘𝑦) → 𝑥 = 𝑦))
95	94	rgen2 3178	. 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)))
97	1, 95, 96	mpbir2an 712	1 ⊢ bits:ℤ–1-1→𝒫 ℕ0

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3887   ∩ cin 3889  𝒫 cpw 4542   class class class wbr 5086   ↾ cres 5626  ⟶wf 6488  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  ωcom 7810   ≈ cen 8883  Fincfn 8886  ℂcc 11027  ℝcr 11028  1c1 11030   − cmin 11368  -cneg 11369  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  bitscbits 16379

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-oi 9418  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-sum 15640  df-dvds 16213  df-bits 16382

This theorem is referenced by:  bitsuz  16434  eulerpartlemmf  34535