Theorem axinfnd 9307

Description: A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)

Assertion

Ref	Expression
axinfnd	⊢ ∃𝑥(𝑦 ∈ 𝑧 → (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))

Proof of Theorem axinfnd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axinfndlem1 9306	. . . . . . 7 ⊢ (∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
2	1	ax-gen 1713	. . . . . 6 ⊢ ∀𝑤(∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
3		nfnae 2306	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		nfnae 2306	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
5	3, 4	nfan 1816	. . . . . . 7 ⊢ Ⅎ𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6		nfnae 2306	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
7		nfnae 2306	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
8	6, 7	nfan 1816	. . . . . . . . 9 ⊢ Ⅎ𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
9		nfcvd 2752	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤)
10		nfcvf 2774	. . . . . . . . . . 11 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
11	10	adantl 481	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧)
12	9, 11	nfeld 2759	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤 ∈ 𝑧)
13	8, 12	nfald 2151	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑥 𝑤 ∈ 𝑧)
14		nfcvf 2774	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥)
16	9, 15	nfeld 2759	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤 ∈ 𝑥)
17		nfnae 2306	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑥
18		nfnae 2306	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑧
19	17, 18	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
20		nfnae 2306	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑥
21		nfnae 2306	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑧
22	20, 21	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
23	11, 15	nfeld 2759	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧 ∈ 𝑥)
24	12, 23	nfand 1814	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
25	22, 24	nfexd 2153	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
26	16, 25	nfimd 1812	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
27	19, 26	nfald 2151	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
28	16, 27	nfand 1814	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
29	8, 28	nfexd 2153	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
30	13, 29	nfimd 1812	. . . . . . 7 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
31		nfcvd 2752	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥𝑤)
32		nfcvf2 2775	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑥𝑦)
33	32	adantr 480	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥𝑦)
34	31, 33	nfeqd 2758	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
35	8, 34	nfan1 2056	. . . . . . . . . 10 ⊢ Ⅎ𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
36		simpr 476	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → 𝑤 = 𝑦)
37	36	eleq1d 2672	. . . . . . . . . 10 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
38	35, 37	albid 2077	. . . . . . . . 9 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥 𝑤 ∈ 𝑧 ↔ ∀𝑥 𝑦 ∈ 𝑧))
39	36	eleq1d 2672	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
40		nfcvd 2752	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑤)
41		nfcvf2 2775	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦)
42	41	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑦)
43	40, 42	nfeqd 2758	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
44	22, 43	nfan1 2056	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
45	37	anbi1d 737	. . . . . . . . . . . . . . . 16 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
46	44, 45	exbid 2078	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
47	39, 46	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
48	47	ex 449	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
49	5, 26, 48	cbvald 2265	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
50	49	adantr 480	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
51	39, 50	anbi12d 743	. . . . . . . . . 10 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
52	35, 51	exbid 2078	. . . . . . . . 9 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
53	38, 52	imbi12d 333	. . . . . . . 8 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) ↔ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))))
54	53	ex 449	. . . . . . 7 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑤 = 𝑦 → ((∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) ↔ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))))
55	5, 30, 54	cbvald 2265	. . . . . 6 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑤(∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) ↔ ∀𝑦(∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))))
56	2, 55	mpbii 222	. . . . 5 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑦(∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
57	56	19.21bi 2047	. . . 4 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
58	57	ex 449	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))))
59		nd1 9288	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
60	59	aecoms 2300	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
61	60	pm2.21d 117	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
62		nd3 9290	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
63	62	pm2.21d 117	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
64	58, 61, 63	pm2.61ii 176	. 2 ⊢ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
65	64	19.35ri 1796	1 ⊢ ∃𝑥(𝑦 ∈ 𝑧 → (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnfc 2738

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-pr 4833  ax-reg 8380  ax-inf 8418

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-nul 3875  df-sn 4126  df-pr 4128

This theorem is referenced by:  zfcndinf  9319  axinfprim  30837