Theorem axpowndlem2 9299

Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.)

Assertion

Ref	Expression
axpowndlem2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))

Distinct variable group:   𝑦,𝑧

Proof of Theorem axpowndlem2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfpow 4770	. . . 4 ⊢ ∃𝑤∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤)
2		19.8a 2039	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑦 → ∃𝑧 𝑤 ∈ 𝑦)
3		sp 2041	. . . . . . . 8 ⊢ (∀𝑦 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑧)
4	2, 3	imim12i 60	. . . . . . 7 ⊢ ((∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧))
5	4	alimi 1730	. . . . . 6 ⊢ (∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → ∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧))
6	5	imim1i 61	. . . . 5 ⊢ ((∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤) → (∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤))
7	6	alimi 1730	. . . 4 ⊢ (∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤) → ∀𝑦(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤))
8	1, 7	eximii 1754	. . 3 ⊢ ∃𝑤∀𝑦(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤)
9		nfnae 2306	. . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
10		nfnae 2306	. . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑧
11	9, 10	nfan 1816	. . . 4 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
12		nfnae 2306	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
13		nfnae 2306	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑧
14	12, 13	nfan 1816	. . . . 5 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
15		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
16		nfnae 2306	. . . . . . . . . 10 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
17		nfcvd 2752	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑤)
18		nfcvf 2774	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
19	17, 18	nfeld 2759	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 ∈ 𝑦)
20	16, 19	nfexd 2153	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∃𝑧 𝑤 ∈ 𝑦)
21	20	adantr 480	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥∃𝑧 𝑤 ∈ 𝑦)
22		nfcvd 2752	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝑤)
23		nfcvf 2774	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝑧)
24	22, 23	nfeld 2759	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑤 ∈ 𝑧)
25	13, 24	nfald 2151	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥∀𝑦 𝑤 ∈ 𝑧)
26	25	adantl 481	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥∀𝑦 𝑤 ∈ 𝑧)
27	21, 26	nfimd 1812	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧))
28	15, 27	nfald 2151	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧))
29	18, 17	nfeld 2759	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑤)
30	29	adantr 480	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 ∈ 𝑤)
31	28, 30	nfimd 1812	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤))
32	14, 31	nfald 2151	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥∀𝑦(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤))
33		nfeqf2 2285	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑤 = 𝑥)
34	33	naecoms 2301	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = 𝑥)
35	34	adantr 480	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑦 𝑤 = 𝑥)
36	14, 35	nfan1 2056	. . . . . 6 ⊢ Ⅎ𝑦((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
37		nfnae 2306	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑧
38		nfeqf2 2285	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑤 = 𝑥)
39	38	naecoms 2301	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑧 𝑤 = 𝑥)
40	37, 39	nfan1 2056	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(¬ ∀𝑥 𝑥 = 𝑧 ∧ 𝑤 = 𝑥)
41		elequ1 1984	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
42	41	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ 𝑤 = 𝑥) → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
43	40, 42	exbid 2078	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ 𝑤 = 𝑥) → (∃𝑧 𝑤 ∈ 𝑦 ↔ ∃𝑧 𝑥 ∈ 𝑦))
44	43	adantll 746	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑧 𝑤 ∈ 𝑦 ↔ ∃𝑧 𝑥 ∈ 𝑦))
45	12, 34	nfan1 2056	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑥)
46		elequ1 1984	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
47	46	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
48	45, 47	albid 2077	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑤 = 𝑥) → (∀𝑦 𝑤 ∈ 𝑧 ↔ ∀𝑦 𝑥 ∈ 𝑧))
49	48	adantlr 747	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦 𝑤 ∈ 𝑧 ↔ ∀𝑦 𝑥 ∈ 𝑧))
50	44, 49	imbi12d 333	. . . . . . . . . 10 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
51	50	ex 449	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧))))
52	11, 27, 51	cbvald 2265	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
53	52	adantr 480	. . . . . . 7 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
54		elequ2 1991	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥))
55	54	adantl 481	. . . . . . 7 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥))
56	53, 55	imbi12d 333	. . . . . 6 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤) ↔ (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
57	36, 56	albid 2077	. . . . 5 ⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
58	57	ex 449	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → (∀𝑦(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
59	11, 32, 58	cbvexd 2266	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤∀𝑦(∀𝑤(∃𝑧 𝑤 ∈ 𝑦 → ∀𝑦 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑤) ↔ ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
60	8, 59	mpbii 222	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
61	60	ex 449	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

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-pow 4769

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740

This theorem is referenced by:  axpowndlem3  9300