Theorem axregnd 10012

Description: A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
axregnd	⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Proof of Theorem axregnd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axregndlem2 10011	. . . 4 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦)))
2		nfnae 2456	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3		nfnae 2456	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑦
4	2, 3	nfan 1900	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
5		nfnae 2456	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
6		nfnae 2456	. . . . . . . 8 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
7	5, 6	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
8		nfcvf 3007	. . . . . . . . . 10 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑥)
9	8	nfcrd 2969	. . . . . . . . 9 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑤 ∈ 𝑥)
10	9	adantr 483	. . . . . . . 8 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑥)
11		nfcvf 3007	. . . . . . . . . . 11 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝑦)
12	11	nfcrd 2969	. . . . . . . . . 10 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑤 ∈ 𝑦)
13	12	nfnd 1858	. . . . . . . . 9 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 ¬ 𝑤 ∈ 𝑦)
14	13	adantl 484	. . . . . . . 8 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 ¬ 𝑤 ∈ 𝑦)
15	10, 14	nfimd 1895	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦))
16		elequ1 2121	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
17		elequ1 2121	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
18	17	notbid 320	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑦))
19	16, 18	imbi12d 347	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
20	19	a1i 11	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
21	7, 15, 20	cbvald 2428	. . . . . 6 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
22	21	anbi2d 630	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 ∈ 𝑦 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
23	4, 22	exbid 2225	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ¬ 𝑤 ∈ 𝑦)) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
24	1, 23	syl5ib 246	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
25	24	ex 415	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))))
26		axregndlem1 10010	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
27	26	aecoms 2450	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
28		19.8a 2180	. . 3 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥 𝑥 ∈ 𝑦)
29		nfae 2455	. . . 4 ⊢ Ⅎ𝑥∀𝑧 𝑧 = 𝑦
30		elirrv 9046	. . . . . . . . 9 ⊢ ¬ 𝑧 ∈ 𝑧
31		elequ2 2129	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑧 ↔ 𝑧 ∈ 𝑦))
32	30, 31	mtbii 328	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ¬ 𝑧 ∈ 𝑦)
33	32	a1d 25	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))
34	33	alimi 1812	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))
35	34	anim2i 618	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 𝑧 = 𝑦) → (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
36	35	expcom 416	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
37	29, 36	eximd 2216	. . 3 ⊢ (∀𝑧 𝑧 = 𝑦 → (∃𝑥 𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
38	28, 37	syl5 34	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
39	25, 27, 38	pm2.61ii 185	1 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

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 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316  ax-reg 9042

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3488  df-dif 3927  df-un 3929  df-nul 4280  df-sn 4554  df-pr 4556

This theorem is referenced by:  zfcndreg  10025  axregprim  32938