Theorem axpowndlem3 9300

Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.)

Assertion

Ref	Expression
axpowndlem3	⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

Distinct variable group:   𝑦,𝑧

Proof of Theorem axpowndlem3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sp 2041	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2	1	con3i 149	. 2 ⊢ (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3		p0ex 4779	. . . . . . . 8 ⊢ {∅} ∈ V
4		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑥 = {∅} → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ {∅}))
5	4	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = {∅} → ((𝑤 = ∅ → 𝑤 ∈ 𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ {∅})))
6	5	albidv 1836	. . . . . . . 8 ⊢ (𝑥 = {∅} → (∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅})))
7	3, 6	spcev 3273	. . . . . . 7 ⊢ (∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅}) → ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
8		0ex 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
9	8	snid 4155	. . . . . . . 8 ⊢ ∅ ∈ {∅}
10		eleq1 2676	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
11	9, 10	mpbiri 247	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝑤 ∈ {∅})
12	7, 11	mpg 1715	. . . . . 6 ⊢ ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥)
13		neq0 3889	. . . . . . . . . 10 ⊢ (¬ 𝑤 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝑤)
14	13	con1bii 345	. . . . . . . . 9 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝑤 ↔ 𝑤 = ∅)
15	14	imbi1i 338	. . . . . . . 8 ⊢ ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ 𝑥))
16	15	albii 1737	. . . . . . 7 ⊢ (∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
17	16	exbii 1764	. . . . . 6 ⊢ (∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
18	12, 17	mpbir 220	. . . . 5 ⊢ ∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥)
19		nfnae 2306	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
20		nfnae 2306	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
21		nfcvf2 2775	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
22		nfcvd 2752	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
23	21, 22	nfeld 2759	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤)
24	19, 23	nfexd 2153	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∃𝑥 𝑥 ∈ 𝑤)
25	24	nfnd 1769	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 ¬ ∃𝑥 𝑥 ∈ 𝑤)
26	22, 21	nfeld 2759	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 ∈ 𝑥)
27	25, 26	nfimd 1812	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥))
28		nfeqf2 2285	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 = 𝑦)
29	19, 28	nfan1 2056	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦)
30		elequ2 1991	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
31	30	adantl 481	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
32	29, 31	exbid 2078	. . . . . . . . . 10 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (∃𝑥 𝑥 ∈ 𝑤 ↔ ∃𝑥 𝑥 ∈ 𝑦))
33	32	notbid 307	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (¬ ∃𝑥 𝑥 ∈ 𝑤 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑦))
34		elequ1 1984	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
35	34	adantl 481	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
36	33, 35	imbi12d 333	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
37	36	ex 449	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥))))
38	20, 27, 37	cbvald 2265	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
39	19, 38	exbid 2078	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
40	18, 39	mpbii 222	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥))
41		nfae 2304	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑧
42		nfae 2304	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑧
43		axc11r 2175	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑥 ¬ 𝑥 ∈ 𝑦))
44		alnex 1697	. . . . . . . . . 10 ⊢ (∀𝑧 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)
45		alnex 1697	. . . . . . . . . 10 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑦)
46	43, 44, 45	3imtr3g 283	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 → ¬ ∃𝑥 𝑥 ∈ 𝑦))
47		nd3 9290	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧)
48	47	pm2.21d 117	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥 ∈ 𝑧 → ¬ ∃𝑥 𝑥 ∈ 𝑦))
49	46, 48	jad 173	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑧 → ((∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → ¬ ∃𝑥 𝑥 ∈ 𝑦))
50	49	spsd 2045	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → ¬ ∃𝑥 𝑥 ∈ 𝑦))
51	50	imim1d 80	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → ((¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
52	42, 51	alimd 2068	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
53	41, 52	eximd 2072	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
54	40, 53	syl5com 31	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
55		axpowndlem2 9299	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
56	54, 55	pm2.61d 169	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
57	2, 56	syl 17	1 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∅c0 3874  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128

This theorem is referenced by:  axpowndlem4  9301