Theorem axpowndlem3 10723

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 2404. (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 2223	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		p0ex 5353	. . . . . . . 8 ⊢ {∅} ∈ V
3		eleq2 2853	. . . . . . . . . 10 ⊢ (𝑥 = {∅} → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ {∅}))
4	3	imbi2d 343	. . . . . . . . 9 ⊢ (𝑥 = {∅} → ((𝑤 = ∅ → 𝑤 ∈ 𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ {∅})))
5	4	albidv 1955	. . . . . . . 8 ⊢ (𝑥 = {∅} → (∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅})))
6	2, 5	spcev 3564	. . . . . . 7 ⊢ (∀𝑤(𝑤 = ∅ → 𝑤 ∈ {∅}) → ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
7		0ex 5267	. . . . . . . . 9 ⊢ ∅ ∈ V
8	7	snid 4624	. . . . . . . 8 ⊢ ∅ ∈ {∅}
9		eleq1 2852	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
10	8, 9	mpbiri 261	. . . . . . 7 ⊢ (𝑤 = ∅ → 𝑤 ∈ {∅})
11	6, 10	mpg 1830	. . . . . 6 ⊢ ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥)
12		neq0 4302	. . . . . . . . . 10 ⊢ (¬ 𝑤 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝑤)
13	12	con1bii 359	. . . . . . . . 9 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝑤 ↔ 𝑤 = ∅)
14	13	imbi1i 352	. . . . . . . 8 ⊢ ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (𝑤 = ∅ → 𝑤 ∈ 𝑥))
15	14	albii 1852	. . . . . . 7 ⊢ (∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
16	15	exbii 1881	. . . . . 6 ⊢ (∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑤(𝑤 = ∅ → 𝑤 ∈ 𝑥))
17	11, 16	mpbir 234	. . . . 5 ⊢ ∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥)
18		nfnae 2466	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19		nfnae 2466	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
20		nfcvf2 2953	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
21		nfcvd 2927	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑤)
22	20, 21	nfeld 2937	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 ∈ 𝑤)
23	18, 22	nfexd 2363	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦∃𝑥 𝑥 ∈ 𝑤)
24	23	nfnd 1891	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 ¬ ∃𝑥 𝑥 ∈ 𝑤)
25	21, 20	nfeld 2937	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 ∈ 𝑥)
26	24, 25	nfimd 1929	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥))
27		nfeqf2 2409	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 = 𝑦)
28	18, 27	nfan1 2240	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦)
29		elequ2 2162	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
30	29	adantl 487	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
31	28, 30	exbid 2263	. . . . . . . . . 10 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (∃𝑥 𝑥 ∈ 𝑤 ↔ ∃𝑥 𝑥 ∈ 𝑦))
32	31	notbid 321	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (¬ ∃𝑥 𝑥 ∈ 𝑤 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑦))
33		elequ1 2154	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
34	33	adantl 487	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
35	32, 34	imbi12d 347	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑦) → ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
36	35	ex 418	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑦 → ((¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ (¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥))))
37	19, 26, 36	cbvald 2439	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
38	18, 37	exbid 2263	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥∀𝑤(¬ ∃𝑥 𝑥 ∈ 𝑤 → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
39	17, 38	mpbii 236	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥))
40		nfae 2465	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑧
41		nfae 2465	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 𝑥 = 𝑧
42		axc11r 2400	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑥 ¬ 𝑥 ∈ 𝑦))
43		alnex 1814	. . . . . . . . . 10 ⊢ (∀𝑧 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑧 𝑥 ∈ 𝑦)
44		alnex 1814	. . . . . . . . . 10 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑥 𝑥 ∈ 𝑦)
45	42, 43, 44	3imtr3g 298	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∃𝑧 𝑥 ∈ 𝑦 → ¬ ∃𝑥 𝑥 ∈ 𝑦))
46		nd3 10713	. . . . . . . . . 10 ⊢ (∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑦 𝑥 ∈ 𝑧)
47	46	pm2.21d 122	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥 ∈ 𝑧 → ¬ ∃𝑥 𝑥 ∈ 𝑦))
48	45, 47	jad 189	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑧 → ((∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → ¬ ∃𝑥 𝑥 ∈ 𝑦))
49	48	spsd 2227	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → ¬ ∃𝑥 𝑥 ∈ 𝑦))
50	49	imim1d 83	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → ((¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
51	41, 50	alimd 2252	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦(¬ ∃𝑥 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
52	40, 51	eximd 2256	. . . 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 32	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
54		axpowndlem2 10722	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
55	53, 54	pm2.61d 181	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
56	1, 55	nsyl5 160	1 ⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∅c0 4281  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-reg 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282  df-pw 4562  df-sn 4589

This theorem is used by:  axpowndlem4  10724