Theorem axinfnd 10600

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 10599	. . . . . . 7 ⊢ (∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
2	1	ax-gen 1789	. . . . . 6 ⊢ ∀𝑤(∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
3		nfnae 2427	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		nfnae 2427	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑧
5	3, 4	nfan 1894	. . . . . . 7 ⊢ Ⅎ𝑦(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
6		nfnae 2427	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥
7		nfnae 2427	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
8	6, 7	nfan 1894	. . . . . . . . 9 ⊢ Ⅎ𝑥(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
9		nfcvd 2898	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑤)
10		nfcvf 2926	. . . . . . . . . . 11 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
11	10	adantl 481	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑧)
12	9, 11	nfeld 2908	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤 ∈ 𝑧)
13	8, 12	nfald 2315	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑥 𝑤 ∈ 𝑧)
14		nfcvf 2926	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦𝑥)
16	9, 15	nfeld 2908	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑤 ∈ 𝑥)
17		nfnae 2427	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑥
18		nfnae 2427	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤 ¬ ∀𝑦 𝑦 = 𝑧
19	17, 18	nfan 1894	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
20		nfnae 2427	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑥
21		nfnae 2427	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 ¬ ∀𝑦 𝑦 = 𝑧
22	20, 21	nfan 1894	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧)
23	11, 15	nfeld 2908	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦 𝑧 ∈ 𝑥)
24	12, 23	nfand 1892	. . . . . . . . . . . . 13 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
25	22, 24	nfexd 2316	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))
26	16, 25	nfimd 1889	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
27	19, 26	nfald 2315	. . . . . . . . . 10 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
28	16, 27	nfand 1892	. . . . . . . . 9 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
29	8, 28	nfexd 2316	. . . . . . . 8 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
30	13, 29	nfimd 1889	. . . . . . 7 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑦(∀𝑥 𝑤 ∈ 𝑧 → ∃𝑥(𝑤 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
31		nfcvd 2898	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥𝑤)
32		nfcvf2 2927	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑥𝑦)
33	32	adantr 480	. . . . . . . . . . . 12 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥𝑦)
34	31, 33	nfeqd 2907	. . . . . . . . . . 11 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑥 𝑤 = 𝑦)
35	8, 34	nfan1 2185	. . . . . . . . . 10 ⊢ Ⅎ𝑥((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
36		simpr 484	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → 𝑤 = 𝑦)
37	36	eleq1d 2812	. . . . . . . . . 10 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
38	35, 37	albid 2207	. . . . . . . . 9 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∀𝑥 𝑤 ∈ 𝑧 ↔ ∀𝑥 𝑦 ∈ 𝑧))
39	36	eleq1d 2812	. . . . . . . . . . 11 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
40		nfcvd 2898	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑤)
41		nfcvf2 2927	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑧𝑦)
42	41	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧𝑦)
43	40, 42	nfeqd 2907	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑤 = 𝑦)
44	22, 43	nfan1 2185	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑧((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦)
45	37	anbi1d 629	. . . . . . . . . . . . . . . 16 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
46	44, 45	exbid 2208	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑤 = 𝑦) → (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
47	39, 46	imbi12d 344	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 2400	. . . . . . . . . . . 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 630	. . . . . . . . . 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 2208	. . . . . . . . 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 344	. . . . . . . 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 412	. . . . . . 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 2400	. . . . . 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 232	. . . . 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 2174	. . . 4 ⊢ ((¬ ∀𝑦 𝑦 = 𝑥 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
58	57	ex 412	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))))
59		nd1 10581	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
60	59	aecoms 2421	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
61	60	pm2.21d 121	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
62		nd3 10583	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
63	62	pm2.21d 121	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))))
64	58, 61, 63	pm2.61ii 183	. 2 ⊢ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
65	64	19.35ri 1874	1 ⊢ ∃𝑥(𝑦 ∈ 𝑧 → (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  ∃wex 1773  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697  ax-sep 5292  ax-pr 5420  ax-reg 9586  ax-inf 9632

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626

This theorem is referenced by:  zfcndinf  10612  axinfprim  35208