Theorem bwth 22015

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 3228	. . . . 5 ⊢ ¬ ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin)
4		r19.29 3216	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin) → ∃𝑏 ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ ¬ (𝐴 ∩ 𝑏) ∈ Fin))
5	3, 4	mto 200	. . . 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 3228	. 2 ⊢ ¬ ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
8		ralnex 3199	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
9		cmptop 22000	. . . . . . 7 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
10		bwt2.1	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ 𝐽
11	10	islp3 21751	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
12	11	3expa 1115	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
13	12	notbid 321	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
14	13	ralbidva 3161	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
15	9, 14	sylan 583	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (∀𝑥 ∈ 𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
16	8, 15	bitr3id 288	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋) → (¬ ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
17		rexanali 3224	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) ↔ ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅))
18		nne 2991	. . . . . . . . . . . 12 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅)
19		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		sneq 4535	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = 𝑥 → {𝑜} = {𝑥})
21	20	difeq2d 4050	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = 𝑥 → (𝐴 ∖ {𝑜}) = (𝐴 ∖ {𝑥}))
22	21	ineq2d 4139	. . . . . . . . . . . . . 14 ⊢ (𝑜 = 𝑥 → (𝑏 ∩ (𝐴 ∖ {𝑜})) = (𝑏 ∩ (𝐴 ∖ {𝑥})))
23	22	eqeq1d 2800	. . . . . . . . . . . . 13 ⊢ (𝑜 = 𝑥 → ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅))
24	19, 23	spcev 3555	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
25	18, 24	sylbi 220	. . . . . . . . . . 11 ⊢ (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
26	25	anim2i 619	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
27	26	reximi 3206	. . . . . . . . 9 ⊢ (∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
28	17, 27	sylbir 238	. . . . . . . 8 ⊢ (¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
29	28	ralimi 3128	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ¬ ∀𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
30	10	cmpcov2 21995	. . . . . . . 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 243	. . . 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 1129	. . 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 4123	. . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → 𝑧 ∈ Fin)
37		sseq2 3941	. . . . . . . . . . . 12 ⊢ (𝑋 = ∪ 𝑧 → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝑧))
38	37	biimpac 482	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 = ∪ 𝑧) → 𝐴 ⊆ ∪ 𝑧)
39		infssuni 8799	. . . . . . . . . . . . 13 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin ∧ 𝐴 ⊆ ∪ 𝑧) → ∃𝑏 ∈ 𝑧 ¬ (𝐴 ∩ 𝑏) ∈ Fin)
40	39	3expa 1115	. . . . . . . . . . . 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 583	. . . . . . . . . 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 471	. . . . . . . 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		0fin 8730	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
47		eleq1 2877	. . . . . . . . . . . 12 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ↔ ∅ ∈ Fin))
48	46, 47	mpbiri 261	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin)
49		snfi 8577	. . . . . . . . . . 11 ⊢ {𝑜} ∈ Fin
50		unfi 8769	. . . . . . . . . . 11 ⊢ (((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ∧ {𝑜} ∈ Fin) → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
51	48, 49, 50	sylancl 589	. . . . . . . . . 10 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
52		ssun1 4099	. . . . . . . . . . . 12 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑜})
53		ssun1 4099	. . . . . . . . . . . . 13 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑜})
54		undif1 4382	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑜}) ∪ {𝑜}) = (𝐴 ∪ {𝑜})
55	53, 54	sseqtrri 3952	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})
56		ss2in 4163	. . . . . . . . . . . 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 4128	. . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59		undir 4203	. . . . . . . . . . 11 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) = ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
60	57, 58, 59	3sstr4i 3958	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})
61		ssfi 8722	. . . . . . . . . 10 ⊢ ((((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin ∧ (𝐴 ∩ 𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})) → (𝐴 ∩ 𝑏) ∈ Fin)
62	51, 60, 61	sylancl 589	. . . . . . . . 9 ⊢ ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
63	62	exlimiv 1931	. . . . . . . 8 ⊢ (∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴 ∩ 𝑏) ∈ Fin)
64	63	ralimi 3128	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑧 ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ∀𝑏 ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin)
65	45, 64	anim12ci 616	. . . . . 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 3233	. . . 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 1127	. . 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 151	1 ⊢ ((𝐽 ∈ Comp ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800  ‘cfv 6324  Fincfn 8492  Topctop 21498  limPtclp 21739  Compccmp 21991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-fin 8496  df-top 21499  df-cld 21624  df-ntr 21625  df-cls 21626  df-lp 21741  df-cmp 21992

This theorem is referenced by:  poimirlem30  35087  fourierdlem42  42791