Theorem axpowndlem4 10619

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 2377. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
axpowndlem4	⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))

Proof of Theorem axpowndlem4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axpowndlem3 10618	. . . . 5 ⊢ (¬ 𝑥 = 𝑤 → ∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
2	1	ax-gen 1795	. . . 4 ⊢ ∀𝑤(¬ 𝑥 = 𝑤 → ∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
3		nfnae 2439	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		nfnae 2439	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
5	3, 4	nfan 1899	. . . . 5 ⊢ Ⅎ𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6		nfcvf 2926	. . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
7	6	adantr 480	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥)
8		nfcvd 2900	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤)
9	7, 8	nfeqd 2910	. . . . . . 7 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥 = 𝑤)
10	9	nfnd 1858	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 ¬ 𝑥 = 𝑤)
11		nfnae 2439	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
12		nfnae 2439	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
13	11, 12	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
14		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
15		nfnae 2439	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑥
16		nfnae 2439	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑧
17	15, 16	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
18	7, 8	nfeld 2911	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥 ∈ 𝑤)
19	17, 18	nfexd 2330	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∃𝑧 𝑥 ∈ 𝑤)
20		nfcvf 2926	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
21	20	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧)
22	7, 21	nfeld 2911	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑥 ∈ 𝑧)
23	14, 22	nfald 2329	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑤 𝑥 ∈ 𝑧)
24	19, 23	nfimd 1894	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧))
25	13, 24	nfald 2329	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧))
26	8, 7	nfeld 2911	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤 ∈ 𝑥)
27	25, 26	nfimd 1894	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
28	14, 27	nfald 2329	. . . . . . 7 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
29	13, 28	nfexd 2330	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥))
30	10, 29	nfimd 1894	. . . . 5 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(¬ 𝑥 = 𝑤 → ∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)))
31		equequ2 2026	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
32	31	notbid 318	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (¬ 𝑥 = 𝑤 ↔ ¬ 𝑥 = 𝑦))
33	32	adantl 481	. . . . . . 7 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (¬ 𝑥 = 𝑤 ↔ ¬ 𝑥 = 𝑦))
34		nfcvd 2900	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥𝑤)
35		nfcvf2 2927	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑥𝑦)
36	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥𝑦)
37	34, 36	nfeqd 2910	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
38	13, 37	nfan1 2201	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
39		nfcvd 2900	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑤)
40		nfcvf2 2927	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦)
41	40	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑦)
42	39, 41	nfeqd 2910	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
43	17, 42	nfan1 2201	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
44		elequ2 2124	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
45	44	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
46	43, 45	exbid 2224	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧 𝑥 ∈ 𝑤 ↔ ∃𝑧 𝑥 ∈ 𝑦))
47		biidd 262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
48	47	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)))
49	5, 22, 48	cbvald 2412	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤 𝑥 ∈ 𝑧 ↔ ∀𝑦 𝑥 ∈ 𝑧))
50	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑤 𝑥 ∈ 𝑧 ↔ ∀𝑦 𝑥 ∈ 𝑧))
51	46, 50	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) ↔ (∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
52	38, 51	albid 2223	. . . . . . . . . . . 12 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) ↔ ∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧)))
53		elequ1 2116	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
54	53	adantl 481	. . . . . . . . . . . 12 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
55	52, 54	imbi12d 344	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
56	55	ex 412	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ (∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
57	5, 27, 56	cbvald 2412	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
58	13, 57	exbid 2224	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
59	58	adantr 480	. . . . . . 7 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
60	33, 59	imbi12d 344	. . . . . 6 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((¬ 𝑥 = 𝑤 → ∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)) ↔ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
61	60	ex 412	. . . . 5 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((¬ 𝑥 = 𝑤 → ∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)) ↔ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))))
62	5, 30, 61	cbvald 2412	. . . 4 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(¬ 𝑥 = 𝑤 → ∃𝑥∀𝑤(∀𝑥(∃𝑧 𝑥 ∈ 𝑤 → ∀𝑤 𝑥 ∈ 𝑧) → 𝑤 ∈ 𝑥)) ↔ ∀𝑦(¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
63	2, 62	mpbii 233	. . 3 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
64	63	19.21bi 2190	. 2 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
65	64	ex 412	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-reg 9611

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-pw 4582  df-sn 4607  df-pr 4609

This theorem is referenced by:  axpownd  10620