Theorem bwth 23397

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		pm3.24 404	. . . . . . 7 ⊢ ¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
2	1	a1i 11	. . . . . 6 ⊢ (𝑏 ∈ 𝑧 → ¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin))
3	2	nrex 3069	. . . . 5 ⊢ ¬ ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
4		r19.29 3104	. . . . 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 3069	. 2 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
8		ralnex 3067	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
9		cmptop 23382	. . . . . . 7 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
10		bwt2.1	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 23133	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
12	11	3expa 1125	. . . . . . . . 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 3162	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
15	9, 14	sylan 587	. . . . . 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 3095	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅))
18		nne 2940	. . . . . . . . . . . 12 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅)
19		vex 3437	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		sneq 4568	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = 𝑥 → {𝑜} = {𝑥})
21	20	difeq2d 4060	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑥 → (𝐴 ∖ {𝑜}) = (𝐴 ∖ {𝑥}))
22	21	ineq2d 4152	. . . . . . . . . . . . . 14 ⊢ (𝑜 = 𝑥 → (𝑏 ∩ (𝐴 ∖ {𝑜})) = (𝑏 ∩ (𝐴 ∖ {𝑥})))
23	22	eqeq1d 2743	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑥 → ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅))
24	19, 23	spcev 3546	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
25	18, 24	sylbi 219	. . . . . . . . . . 11 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
26	25	anim2i 624	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
27	26	reximi 3079	. . . . . . . . 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 3078	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
30	10	cmpcov2 23377	. . . . . . . 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 414	. . . . . . 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 482	. . . . 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 1139	. . 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 4134	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → 𝑧 ∈ Fin)
37		sseq2 3943	. . . . . . . . . . . 12 ⊢ (𝑋 = ∪ 𝑧 → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝑧))
38	37	biimpac 480	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) → 𝐴 ⊆ ∪ 𝑧)
39		infssuni 9250	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
40	39	3expa 1125	. . . . . . . . . . . 12 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin) ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
41	40	ancoms 460	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ∪ 𝑧 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
42	38, 41	sylan 587	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
43	42	an42s 668	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑧 ∈ Fin ∧ 𝑋 = ∪ 𝑧)) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
44	43	anassrs 469	. . . . . . . 8 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ Fin) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
45	36, 44	sylanl2 688	. . . . . . 7 ⊢ ((((𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
46		0fi 8983	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
47		eleq1 2829	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ↔ ∅ ∈ Fin))
48	46, 47	mpbiri 260	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin)
49		snfi 8984	. . . . . . . . . . 11 ⊢ {𝑜} ∈ Fin
50		unfi 9099	. . . . . . . . . . 11 ⊢ (((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ∧ {𝑜} ∈ Fin) → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
51	48, 49, 50	sylancl 593	. . . . . . . . . 10 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
52		ssun1 4110	. . . . . . . . . . . 12 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑜})
53		ssun1 4110	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑜})
54		undif1 4407	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑜}) ∪ {𝑜}) = (𝐴 ∪ {𝑜})
55	53, 54	sseqtrri 3966	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})
56		ss2in 4176	. . . . . . . . . . . 12 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑜}) ∧ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})) → (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜})))
57	52, 55, 56	mp2an 699	. . . . . . . . . . 11 ⊢ (𝑏 ∩ 𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
58		incom 4141	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59		undir 4218	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) = ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
60	57, 58, 59	3sstr4i 3968	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})
61		ssfi 9101	. . . . . . . . . 10 ⊢ ((((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin ∧ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})) → (𝐴 ∩ 𝑏) ∈ Fin)
62	51, 60, 61	sylancl 593	. . . . . . . . 9 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
63	62	exlimiv 1938	. . . . . . . 8 ⊢ (∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
64	63	ralimi 3078	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin)
65	45, 64	anim12ci 621	. . . . . 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 459	. . . . 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 3154	. . . 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 1137	. . 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 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065   ∖ cdif 3882   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  𝒫 cpw 4532  {csn 4558  ∪ cuni 4841  ‘cfv 6489  Fincfn 8887  Topctop 22880  limPtclp 23121  Compccmp 23373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-en 8888  df-fin 8891  df-top 22881  df-cld 23006  df-ntr 23007  df-cls 23008  df-lp 23123  df-cmp 23374

This theorem is referenced by:  poimirlem30  38032  fourierdlem42  46606