Theorem zfcndac 10043

Description: Axiom of Choice ax-ac 9883, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
zfcndac	⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡

Proof of Theorem zfcndac

Step	Hyp	Ref	Expression
1		axacnd 10036	. . 3 ⊢ ∃𝑦∀𝑧∀𝑤(∀𝑦(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥))
2		19.3v 1986	. . . . . 6 ⊢ (∀𝑦(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
3	2	imbi1i 352	. . . . 5 ⊢ ((∀𝑦(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)) ↔ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
4	3	2albii 1821	. . . 4 ⊢ (∀𝑧∀𝑤(∀𝑦(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)) ↔ ∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
5	4	exbii 1848	. . 3 ⊢ (∃𝑦∀𝑧∀𝑤(∀𝑦(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)) ↔ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
6	1, 5	mpbi 232	. 2 ⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥))
7		equequ2 2033	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → (𝑢 = 𝑣 ↔ 𝑢 = 𝑥))
8	7	bibi2d 345	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → ((∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) ↔ (∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑥)))
9		elequ2 2129	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ 𝑥))
10	9	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ↔ (𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
11		elequ2 2129	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑥))
12		elequ1 2121	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑡 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
13	11, 12	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → ((𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦) ↔ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)))
14	10, 13	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑥 → (((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦))))
15	14	cbvexvw 2044	. . . . . . . . . 10 ⊢ (∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ ∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)))
16	15	bibi1i 341	. . . . . . . . 9 ⊢ ((∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑥) ↔ (∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑢 = 𝑥))
17	8, 16	syl6bb 289	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → ((∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) ↔ (∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑢 = 𝑥)))
18	17	albidv 1921	. . . . . . 7 ⊢ (𝑣 = 𝑥 → (∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) ↔ ∀𝑢(∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑢 = 𝑥)))
19		elequ1 2121	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
20	19	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)))
21		elequ1 2121	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
22	21	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → ((𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)))
23	20, 22	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦))))
24	23	exbidv 1922	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ ∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦))))
25		equequ1 2032	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑥 ↔ 𝑧 = 𝑥))
26	24, 25	bibi12d 348	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → ((∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑢 = 𝑥) ↔ (∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
27	26	cbvalvw 2043	. . . . . . 7 ⊢ (∀𝑢(∃𝑥((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑢 = 𝑥) ↔ ∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥))
28	18, 27	syl6bb 289	. . . . . 6 ⊢ (𝑣 = 𝑥 → (∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) ↔ ∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
29	28	cbvexvw 2044	. . . . 5 ⊢ (∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) ↔ ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥))
30	29	imbi2i 338	. . . 4 ⊢ (((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) ↔ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
31	30	2albii 1821	. . 3 ⊢ (∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) ↔ ∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
32	31	exbii 1848	. 2 ⊢ (∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) ↔ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑥∀𝑧(∃𝑥((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) ↔ 𝑧 = 𝑥)))
33	6, 32	mpbir 233	1 ⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-reg 9058  ax-ac 9883

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-eprel 5467  df-fr 5516

This theorem is referenced by: (None)