Theorem bwth 23304

Description: The glorious Bolzano-Weierstrass theorem. The first general topology theorem ever proved. The first mention of this theorem can be found in a course by Weierstrass from 1865. In his course Weierstrass called it a lemma. He didn't know how famous this theorem would be. He used a Euclidean space instead of a general compact space. And he was not aware of the Heine-Borel property. But the concepts of neighborhood and limit point were already there although not precisely defined. Cantor was one of his students. He published and used the theorem in an article from 1872. The rest of the general topology followed from that. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) Revised by BL to significantly shorten the proof and avoid infinity, regularity, and choice. (Revised by Brendan Leahy, 26-Dec-2018.)

Hypothesis

Ref	Expression
bwt2.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
bwth	⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋

Proof of Theorem bwth

Dummy variables 𝑜 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 402	. . . . . . 7 ⊢ ¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
2	1	a1i 11	. . . . . 6 ⊢ (𝑏 ∈ 𝑧 → ¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin))
3	2	nrex 3058	. . . . 5 ⊢ ¬ ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
4		r19.29 3095	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin) → ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin))
5	3, 4	mto 197	. . . 4 ⊢ ¬ (∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
6	5	a1i 11	. . 3 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → ¬ (∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin))
7	6	nrex 3058	. 2 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
8		ralnex 3056	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
9		cmptop 23289	. . . . . . 7 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
10		bwt2.1	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 23040	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
12	11	3expa 1118	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
13	12	notbid 318	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
14	13	ralbidva 3155	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
15	9, 14	sylan 580	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
16	8, 15	bitr3id 285	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
17		rexanali 3085	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅))
18		nne 2930	. . . . . . . . . . . 12 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅)
19		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		sneq 4602	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = 𝑥 → {𝑜} = {𝑥})
21	20	difeq2d 4092	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑥 → (𝐴 ∖ {𝑜}) = (𝐴 ∖ {𝑥}))
22	21	ineq2d 4186	. . . . . . . . . . . . . 14 ⊢ (𝑜 = 𝑥 → (𝑏 ∩ (𝐴 ∖ {𝑜})) = (𝑏 ∩ (𝐴 ∖ {𝑥})))
23	22	eqeq1d 2732	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑥 → ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅))
24	19, 23	spcev 3575	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
25	18, 24	sylbi 217	. . . . . . . . . . 11 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
26	25	anim2i 617	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
27	26	reximi 3068	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
28	17, 27	sylbir 235	. . . . . . . 8 ⊢ (¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
29	28	ralimi 3067	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
30	10	cmpcov2 23284	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
31	30	ex 412	. . . . . . 7 ⊢ (𝐽 ∈ Comp → (∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
32	29, 31	syl5 34	. . . . . 6 ⊢ (𝐽 ∈ Comp → (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
33	32	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
34	16, 33	sylbid 240	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
35	34	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
36		elinel2 4168	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → 𝑧 ∈ Fin)
37		sseq2 3976	. . . . . . . . . . . 12 ⊢ (𝑋 = ∪ 𝑧 → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝑧))
38	37	biimpac 478	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) → 𝐴 ⊆ ∪ 𝑧)
39		infssuni 9304	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
40	39	3expa 1118	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin) ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
41	40	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ∪ 𝑧 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
42	38, 41	sylan 580	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
43	42	an42s 661	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑧 ∈ Fin ∧ 𝑋 = ∪ 𝑧)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
44	43	anassrs 467	. . . . . . . 8 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ Fin) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
45	36, 44	sylanl2 681	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
46		0fi 9016	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
47		eleq1 2817	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ↔ ∅ ∈ Fin))
48	46, 47	mpbiri 258	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin)
49		snfi 9017	. . . . . . . . . . 11 ⊢ {𝑜} ∈ Fin
50		unfi 9141	. . . . . . . . . . 11 ⊢ (((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ∧ {𝑜} ∈ Fin) → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
51	48, 49, 50	sylancl 586	. . . . . . . . . 10 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
52		ssun1 4144	. . . . . . . . . . . 12 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑜})
53		ssun1 4144	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑜})
54		undif1 4442	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑜}) ∪ {𝑜}) = (𝐴 ∪ {𝑜})
55	53, 54	sseqtrri 3999	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})
56		ss2in 4211	. . . . . . . . . . . 12 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑜}) ∧ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})) → (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜})))
57	52, 55, 56	mp2an 692	. . . . . . . . . . 11 ⊢ (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
58		incom 4175	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59		undir 4253	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) = ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
60	57, 58, 59	3sstr4i 4001	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})
61		ssfi 9143	. . . . . . . . . 10 ⊢ ((((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin ∧ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})) → (𝐴 ∩ 𝑏) ∈ Fin)
62	51, 60, 61	sylancl 586	. . . . . . . . 9 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
63	62	exlimiv 1930	. . . . . . . 8 ⊢ (∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
64	63	ralimi 3067	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin)
65	45, 64	anim12ci 614	. . . . . 6 ⊢ (((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = ∪ 𝑧) ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → (∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin))
66	65	expl 457	. . . . 5 ⊢ (((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → (∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)))
67	66	reximdva 3147	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)))
68	67	3adant1 1130	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑧 ∧ ∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)))
69	35, 68	syld 47	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)))
70	7, 69	mt3i 149	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874  ‘cfv 6514  Fincfn 8921  Topctop 22787  limPtclp 23028  Compccmp 23280

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-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-1o 8437  df-en 8922  df-fin 8925  df-top 22788  df-cld 22913  df-ntr 22914  df-cls 22915  df-lp 23030  df-cmp 23281

This theorem is referenced by:  poimirlem30  37651  fourierdlem42  46154