Theorem axregndlem2 10014

Description: Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
axregndlem2	⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Distinct variable group:   𝑦,𝑧

Proof of Theorem axregndlem2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axreg2 9041	. . . . . 6 ⊢ (𝑤 ∈ 𝑦 → ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)))
2	1	ax-gen 1797	. . . . 5 ⊢ ∀𝑤(𝑤 ∈ 𝑦 → ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)))
3		nfnae 2445	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4		nfnae 2445	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑧
5	3, 4	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
6		nfcvd 2956	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤)
7		nfcvf 2981	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
8	7	adantr 484	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦)
9	6, 8	nfeld 2966	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤 ∈ 𝑦)
10		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
11		nfnae 2445	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
12		nfnae 2445	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑧
13	11, 12	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
14		nfcvf 2981	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝑧)
15	14	adantl 485	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧)
16	15, 6	nfeld 2966	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧 ∈ 𝑤)
17	15, 8	nfeld 2966	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧 ∈ 𝑦)
18	17	nfnd 1859	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 ¬ 𝑧 ∈ 𝑦)
19	16, 18	nfimd 1895	. . . . . . . . . 10 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦))
20	13, 19	nfald 2336	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦))
21	9, 20	nfand 1898	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)))
22	10, 21	nfexd 2337	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)))
23	9, 22	nfimd 1895	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤 ∈ 𝑦 → ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦))))
24		simpr 488	. . . . . . . . 9 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → 𝑤 = 𝑥)
25	24	eleq1d 2874	. . . . . . . 8 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
26		nfcvd 2956	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧𝑤)
27		nfcvf2 2982	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑧𝑥)
28	27	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧𝑥)
29	26, 28	nfeqd 2965	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑥)
30	13, 29	nfan1 2198	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
31	24	eleq2d 2875	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
32	31	imbi1d 345	. . . . . . . . . . . . 13 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
33	30, 32	albid 2222	. . . . . . . . . . . 12 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
34	25, 33	anbi12d 633	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
35	34	ex 416	. . . . . . . . . 10 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))))
36	5, 21, 35	cbvexd 2418	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
37	36	adantr 484	. . . . . . . 8 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦)) ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
38	25, 37	imbi12d 348	. . . . . . 7 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((𝑤 ∈ 𝑦 → ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦))) ↔ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))))
39	38	ex 416	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤 ∈ 𝑦 → ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦))) ↔ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))))
40	5, 23, 39	cbvald 2417	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(𝑤 ∈ 𝑦 → ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑤 → ¬ 𝑧 ∈ 𝑦))) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))))
41	2, 40	mpbii 236	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
42	41	19.21bi 2186	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
43	42	ex 416	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))))
44		elirrv 9044	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
45		elequ2 2126	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑥 ∈ 𝑦))
46	44, 45	mtbii 329	. . . 4 ⊢ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
47	46	sps 2182	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝑦)
48	47	pm2.21d 121	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
49		axregndlem1 10013	. 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
50	43, 48, 49	pm2.61ii 186	1 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnfc 2936

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-reg 9040

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528

This theorem is referenced by:  axregnd  10015