Theorem axpowndlem3 10286

Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.)

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 2178	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		p0ex 5302	. . . . . . . 8 ⊢ {∅} ∈ V
3		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑥 = {∅} → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ {∅}))
4	3	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = {∅} → ((𝑤 = ∅ → 𝑤 ∈ 𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ {∅})))
5	4	albidv 1924	. . . . . . . 8 ⊢ (𝑥 = {∅} → (∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅})))
6	2, 5	spcev 3535	. . . . . . 7 ⊢ (∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅}) → ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
7		0ex 5226	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	snid 4594	. . . . . . . 8 ⊢ ∅ ∈ {∅}
9		eleq1 2826	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
10	8, 9	mpbiri 257	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝑤 ∈ {∅})
11	6, 10	mpg 1801	. . . . . 6 ⊢ ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥)
12		neq0 4276	. . . . . . . . . 10 ⊢ (¬ 𝑤 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝑤)
13	12	con1bii 356	. . . . . . . . 9 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝑤 ↔ 𝑤 = ∅)
14	13	imbi1i 349	. . . . . . . 8 ⊢ ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ 𝑥))
15	14	albii 1823	. . . . . . 7 ⊢ (∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
16	15	exbii 1851	. . . . . 6 ⊢ (∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
17	11, 16	mpbir 230	. . . . 5 ⊢ ∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥)
18		nfnae 2434	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		nfnae 2434	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
20		nfcvf2 2936	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
21		nfcvd 2907	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
22	20, 21	nfeld 2917	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤)
23	18, 22	nfexd 2327	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∃𝑥 𝑥 ∈ 𝑤)
24	23	nfnd 1862	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 ¬ ∃𝑥 𝑥 ∈ 𝑤)
25	21, 20	nfeld 2917	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 ∈ 𝑥)
26	24, 25	nfimd 1898	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥))
27		nfeqf2 2377	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 = 𝑦)
28	18, 27	nfan1 2196	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦)
29		elequ2 2123	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
30	29	adantl 481	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
31	28, 30	exbid 2219	. . . . . . . . . 10 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (∃𝑥 𝑥 ∈ 𝑤 ↔ ∃𝑥 𝑥 ∈ 𝑦))
32	31	notbid 317	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (¬ ∃𝑥 𝑥 ∈ 𝑤 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑦))
33		elequ1 2115	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
34	33	adantl 481	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
35	32, 34	imbi12d 344	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
36	35	ex 412	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥))))
37	19, 26, 36	cbvald 2407	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
38	18, 37	exbid 2219	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
39	17, 38	mpbii 232	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥))
40		nfae 2433	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑧
41		nfae 2433	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑧
42		axc11r 2366	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑥 ¬ 𝑥 ∈ 𝑦))
43		alnex 1785	. . . . . . . . . 10 ⊢ (∀𝑧 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)
44		alnex 1785	. . . . . . . . . 10 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑦)
45	42, 43, 44	3imtr3g 294	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 → ¬ ∃𝑥 𝑥 ∈ 𝑦))
46		nd3 10276	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧)
47	46	pm2.21d 121	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥 ∈ 𝑧 → ¬ ∃𝑥 𝑥 ∈ 𝑦))
48	45, 47	jad 187	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑧 → ((∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → ¬ ∃𝑥 𝑥 ∈ 𝑦))
49	48	spsd 2182	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → ¬ ∃𝑥 𝑥 ∈ 𝑦))
50	49	imim1d 82	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → ((¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
51	41, 50	alimd 2208	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
52	40, 51	eximd 2212	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → (∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
53	39, 52	syl5com 31	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
54		axpowndlem2 10285	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
55	53, 54	pm2.61d 179	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
56	1, 55	nsyl5 159	1 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∅c0 4253  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561

This theorem is referenced by:  axpowndlem4  10287