Theorem bwth 23439

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 3080	. . . . 5 ⊢ ¬ ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
4		r19.29 3120	. . . . 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 3080	. 2 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
8		ralnex 3078	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
9		cmptop 23424	. . . . . . 7 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
10		bwt2.1	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 23175	. . . . . . . . . 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 3182	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
15	9, 14	sylan 579	. . . . . 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 3108	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅))
18		nne 2950	. . . . . . . . . . . 12 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅)
19		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		sneq 4658	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = 𝑥 → {𝑜} = {𝑥})
21	20	difeq2d 4149	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑥 → (𝐴 ∖ {𝑜}) = (𝐴 ∖ {𝑥}))
22	21	ineq2d 4241	. . . . . . . . . . . . . 14 ⊢ (𝑜 = 𝑥 → (𝑏 ∩ (𝐴 ∖ {𝑜})) = (𝑏 ∩ (𝐴 ∖ {𝑥})))
23	22	eqeq1d 2742	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑥 → ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅))
24	19, 23	spcev 3619	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
25	18, 24	sylbi 217	. . . . . . . . . . 11 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
26	25	anim2i 616	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
27	26	reximi 3090	. . . . . . . . 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 3089	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
30	10	cmpcov2 23419	. . . . . . . 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 4225	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → 𝑧 ∈ Fin)
37		sseq2 4035	. . . . . . . . . . . 12 ⊢ (𝑋 = ∪ 𝑧 → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝑧))
38	37	biimpac 478	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) → 𝐴 ⊆ ∪ 𝑧)
39		infssuni 9414	. . . . . . . . . . . . 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 579	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
43	42	an42s 660	. . . . . . . . 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 680	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
46		0fi 9108	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
47		eleq1 2832	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ↔ ∅ ∈ Fin))
48	46, 47	mpbiri 258	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin)
49		snfi 9109	. . . . . . . . . . 11 ⊢ {𝑜} ∈ Fin
50		unfi 9238	. . . . . . . . . . 11 ⊢ (((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ∧ {𝑜} ∈ Fin) → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
51	48, 49, 50	sylancl 585	. . . . . . . . . 10 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
52		ssun1 4201	. . . . . . . . . . . 12 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑜})
53		ssun1 4201	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑜})
54		undif1 4499	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑜}) ∪ {𝑜}) = (𝐴 ∪ {𝑜})
55	53, 54	sseqtrri 4046	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})
56		ss2in 4266	. . . . . . . . . . . 12 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑜}) ∧ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})) → (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜})))
57	52, 55, 56	mp2an 691	. . . . . . . . . . 11 ⊢ (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
58		incom 4230	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59		undir 4306	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) = ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
60	57, 58, 59	3sstr4i 4052	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})
61		ssfi 9240	. . . . . . . . . 10 ⊢ ((((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin ∧ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})) → (𝐴 ∩ 𝑏) ∈ Fin)
62	51, 60, 61	sylancl 585	. . . . . . . . 9 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
63	62	exlimiv 1929	. . . . . . . 8 ⊢ (∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
64	63	ralimi 3089	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin)
65	45, 64	anim12ci 613	. . . . . 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 3174	. . . 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 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931  ‘cfv 6573  Fincfn 9003  Topctop 22920  limPtclp 23163  Compccmp 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-fin 9007  df-top 22921  df-cld 23048  df-ntr 23049  df-cls 23050  df-lp 23165  df-cmp 23416

This theorem is referenced by:  poimirlem30  37610  fourierdlem42  46070