Theorem bwth 23472

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		pm3.24 406	. . . . . . 7 ⊢ ¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
2	1	a1i 11	. . . . . 6 ⊢ (𝑏 ∈ 𝑧 → ¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin))
3	2	nrex 3092	. . . . 5 ⊢ ¬ ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
4		r19.29 3127	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin) → ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin))
5	3, 4	mto 199	. . . 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 3092	. 2 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
8		ralnex 3090	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
9		cmptop 23457	. . . . . . 7 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
10		bwt2.1	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 23208	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
12	11	3expa 1132	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
13	12	notbid 320	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
14	13	ralbidva 3185	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
15	9, 14	sylan 589	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
16	8, 15	bitr3id 287	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
17		rexanali 3118	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅))
18		nne 2963	. . . . . . . . . . . 12 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅)
19		vex 3460	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		sneq 4594	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = 𝑥 → {𝑜} = {𝑥})
21	20	difeq2d 4082	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑥 → (𝐴 ∖ {𝑜}) = (𝐴 ∖ {𝑥}))
22	21	ineq2d 4174	. . . . . . . . . . . . . 14 ⊢ (𝑜 = 𝑥 → (𝑏 ∩ (𝐴 ∖ {𝑜})) = (𝑏 ∩ (𝐴 ∖ {𝑥})))
23	22	eqeq1d 2766	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑥 → ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅))
24	19, 23	spcev 3567	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
25	18, 24	sylbi 219	. . . . . . . . . . 11 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
26	25	anim2i 626	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
27	26	reximi 3102	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
28	17, 27	sylbir 237	. . . . . . . 8 ⊢ (¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
29	28	ralimi 3101	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
30	10	cmpcov2 23452	. . . . . . . 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 416	. . . . . . 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 484	. . . . 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 242	. . . 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 1146	. . 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 4156	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → 𝑧 ∈ Fin)
37		sseq2 3964	. . . . . . . . . . . 12 ⊢ (𝑋 = ∪ 𝑧 → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝑧))
38	37	biimpac 482	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) → 𝐴 ⊆ ∪ 𝑧)
39		infssuni 9291	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
40	39	3expa 1132	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin) ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
41	40	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ∪ 𝑧 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
42	38, 41	sylan 589	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
43	42	an42s 671	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑧 ∈ Fin ∧ 𝑋 = ∪ 𝑧)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
44	43	anassrs 471	. . . . . . . 8 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ Fin) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
45	36, 44	sylanl2 691	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
46		0fi 9025	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
47		eleq1 2852	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ↔ ∅ ∈ Fin))
48	46, 47	mpbiri 260	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin)
49		snfi 9026	. . . . . . . . . . 11 ⊢ {𝑜} ∈ Fin
50		unfi 9141	. . . . . . . . . . 11 ⊢ (((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ∧ {𝑜} ∈ Fin) → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
51	48, 49, 50	sylancl 595	. . . . . . . . . 10 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
52		ssun1 4132	. . . . . . . . . . . 12 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑜})
53		ssun1 4132	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑜})
54		undif1 4432	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑜}) ∪ {𝑜}) = (𝐴 ∪ {𝑜})
55	53, 54	sseqtrri 3987	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})
56		ss2in 4198	. . . . . . . . . . . 12 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑜}) ∧ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})) → (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜})))
57	52, 55, 56	mp2an 702	. . . . . . . . . . 11 ⊢ (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
58		incom 4163	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59		undir 4241	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) = ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
60	57, 58, 59	3sstr4i 3989	. . . . . . . . . 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 595	. . . . . . . . 9 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
63	62	exlimiv 1952	. . . . . . . 8 ⊢ (∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
64	63	ralimi 3101	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin)
65	45, 64	anim12ci 623	. . . . . 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 461	. . . . 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 3177	. . . 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 1144	. . 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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  ‘cfv 6523  Fincfn 8929  Topctop 22955  limPtclp 23196  Compccmp 23448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-om 7849  df-1o 8439  df-en 8930  df-fin 8933  df-top 22956  df-cld 23081  df-ntr 23082  df-cls 23083  df-lp 23198  df-cmp 23449

This theorem is referenced by:  poimirlem30  38154  fourierdlem42  46728