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Theorem bwth 22018
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 ∧ 𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥𝑋 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.
StepHypRef Expression
1 pm3.24 406 . . . . . . 7 ¬ ((𝐴𝑏) ∈ Fin ∧ ¬ (𝐴𝑏) ∈ Fin)
21a1i 11 . . . . . 6 (𝑏𝑧 → ¬ ((𝐴𝑏) ∈ Fin ∧ ¬ (𝐴𝑏) ∈ Fin))
32nrex 3261 . . . . 5 ¬ ∃𝑏𝑧 ((𝐴𝑏) ∈ Fin ∧ ¬ (𝐴𝑏) ∈ Fin)
4 r19.29 3248 . . . . 5 ((∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin) → ∃𝑏𝑧 ((𝐴𝑏) ∈ Fin ∧ ¬ (𝐴𝑏) ∈ Fin))
53, 4mto 200 . . . 4 ¬ (∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
65a1i 11 . . 3 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → ¬ (∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin))
76nrex 3261 . 2 ¬ ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
8 ralnex 3230 . . . . . 6 (∀𝑥𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
9 cmptop 22003 . . . . . . 7 (𝐽 ∈ Comp → 𝐽 ∈ Top)
10 bwt2.1 . . . . . . . . . . 11 𝑋 = 𝐽
1110islp3 21754 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋𝑥𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
12113expa 1115 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
1312notbid 321 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ 𝑥𝑋) → (¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
1413ralbidva 3191 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (∀𝑥𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥𝑋 ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
159, 14sylan 583 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐴𝑋) → (∀𝑥𝑋 ¬ 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥𝑋 ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
168, 15bitr3id 288 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐴𝑋) → (¬ ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑥𝑋 ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅)))
17 rexanali 3257 . . . . . . . . 9 (∃𝑏𝐽 (𝑥𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) ↔ ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅))
18 nne 3018 . . . . . . . . . . . 12 (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅)
19 vex 3483 . . . . . . . . . . . . 13 𝑥 ∈ V
20 sneq 4560 . . . . . . . . . . . . . . . 16 (𝑜 = 𝑥 → {𝑜} = {𝑥})
2120difeq2d 4085 . . . . . . . . . . . . . . 15 (𝑜 = 𝑥 → (𝐴 ∖ {𝑜}) = (𝐴 ∖ {𝑥}))
2221ineq2d 4174 . . . . . . . . . . . . . 14 (𝑜 = 𝑥 → (𝑏 ∩ (𝐴 ∖ {𝑜})) = (𝑏 ∩ (𝐴 ∖ {𝑥})))
2322eqeq1d 2826 . . . . . . . . . . . . 13 (𝑜 = 𝑥 → ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ ↔ (𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅))
2419, 23spcev 3593 . . . . . . . . . . . 12 ((𝑏 ∩ (𝐴 ∖ {𝑥})) = ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
2518, 24sylbi 220 . . . . . . . . . . 11 (¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅ → ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)
2625anim2i 619 . . . . . . . . . 10 ((𝑥𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → (𝑥𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
2726reximi 3237 . . . . . . . . 9 (∃𝑏𝐽 (𝑥𝑏 ∧ ¬ (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏𝐽 (𝑥𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
2817, 27sylbir 238 . . . . . . . 8 (¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑏𝐽 (𝑥𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
2928ralimi 3155 . . . . . . 7 (∀𝑥𝑋 ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∀𝑥𝑋𝑏𝐽 (𝑥𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
3010cmpcov2 21998 . . . . . . . 8 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑏𝐽 (𝑥𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅))
3130ex 416 . . . . . . 7 (𝐽 ∈ Comp → (∀𝑥𝑋𝑏𝐽 (𝑥𝑏 ∧ ∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
3229, 31syl5 34 . . . . . 6 (𝐽 ∈ Comp → (∀𝑥𝑋 ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
3332adantr 484 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐴𝑋) → (∀𝑥𝑋 ¬ ∀𝑏𝐽 (𝑥𝑏 → (𝑏 ∩ (𝐴 ∖ {𝑥})) ≠ ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
3416, 33sylbid 243 . . . 4 ((𝐽 ∈ Comp ∧ 𝐴𝑋) → (¬ ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
35343adant3 1129 . . 3 ((𝐽 ∈ Comp ∧ 𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → (¬ ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅)))
36 elinel2 4158 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) → 𝑧 ∈ Fin)
37 sseq2 3979 . . . . . . . . . . . 12 (𝑋 = 𝑧 → (𝐴𝑋𝐴 𝑧))
3837biimpac 482 . . . . . . . . . . 11 ((𝐴𝑋𝑋 = 𝑧) → 𝐴 𝑧)
39 infssuni 8812 . . . . . . . . . . . . 13 ((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin ∧ 𝐴 𝑧) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
40393expa 1115 . . . . . . . . . . . 12 (((¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin) ∧ 𝐴 𝑧) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
4140ancoms 462 . . . . . . . . . . 11 ((𝐴 𝑧 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
4238, 41sylan 583 . . . . . . . . . 10 (((𝐴𝑋𝑋 = 𝑧) ∧ (¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
4342an42s 660 . . . . . . . . 9 (((𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑧 ∈ Fin ∧ 𝑋 = 𝑧)) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
4443anassrs 471 . . . . . . . 8 ((((𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ Fin) ∧ 𝑋 = 𝑧) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
4536, 44sylanl2 680 . . . . . . 7 ((((𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = 𝑧) → ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)
46 0fin 8743 . . . . . . . . . . . 12 ∅ ∈ Fin
47 eleq1 2903 . . . . . . . . . . . 12 ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ↔ ∅ ∈ Fin))
4846, 47mpbiri 261 . . . . . . . . . . 11 ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin)
49 snfi 8590 . . . . . . . . . . 11 {𝑜} ∈ Fin
50 unfi 8782 . . . . . . . . . . 11 (((𝑏 ∩ (𝐴 ∖ {𝑜})) ∈ Fin ∧ {𝑜} ∈ Fin) → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
5148, 49, 50sylancl 589 . . . . . . . . . 10 ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin)
52 ssun1 4134 . . . . . . . . . . . 12 𝑏 ⊆ (𝑏 ∪ {𝑜})
53 ssun1 4134 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑜})
54 undif1 4407 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑜}) ∪ {𝑜}) = (𝐴 ∪ {𝑜})
5553, 54sseqtrri 3990 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})
56 ss2in 4198 . . . . . . . . . . . 12 ((𝑏 ⊆ (𝑏 ∪ {𝑜}) ∧ 𝐴 ⊆ ((𝐴 ∖ {𝑜}) ∪ {𝑜})) → (𝑏𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜})))
5752, 55, 56mp2an 691 . . . . . . . . . . 11 (𝑏𝐴) ⊆ ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
58 incom 4163 . . . . . . . . . . 11 (𝐴𝑏) = (𝑏𝐴)
59 undir 4238 . . . . . . . . . . 11 ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) = ((𝑏 ∪ {𝑜}) ∩ ((𝐴 ∖ {𝑜}) ∪ {𝑜}))
6057, 58, 593sstr4i 3996 . . . . . . . . . 10 (𝐴𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})
61 ssfi 8735 . . . . . . . . . 10 ((((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜}) ∈ Fin ∧ (𝐴𝑏) ⊆ ((𝑏 ∩ (𝐴 ∖ {𝑜})) ∪ {𝑜})) → (𝐴𝑏) ∈ Fin)
6251, 60, 61sylancl 589 . . . . . . . . 9 ((𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴𝑏) ∈ Fin)
6362exlimiv 1932 . . . . . . . 8 (∃𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → (𝐴𝑏) ∈ Fin)
6463ralimi 3155 . . . . . . 7 (∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅ → ∀𝑏𝑧 (𝐴𝑏) ∈ Fin)
6545, 64anim12ci 616 . . . . . 6 (((((𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = 𝑧) ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → (∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin))
6665expl 461 . . . . 5 (((𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → (∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)))
6766reximdva 3266 . . . 4 ((𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)))
68673adant1 1127 . . 3 ((𝐽 ∈ Comp ∧ 𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑧 ∧ ∀𝑏𝑧𝑜(𝑏 ∩ (𝐴 ∖ {𝑜})) = ∅) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)))
6935, 68syld 47 . 2 ((𝐽 ∈ Comp ∧ 𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → (¬ ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴) → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)(∀𝑏𝑧 (𝐴𝑏) ∈ Fin ∧ ∃𝑏𝑧 ¬ (𝐴𝑏) ∈ Fin)))
707, 69mt3i 151 1 ((𝐽 ∈ Comp ∧ 𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → ∃𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wne 3014  wral 3133  wrex 3134  cdif 3916  cun 3917  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   cuni 4824  cfv 6343  Fincfn 8505  Topctop 21501  limPtclp 21742  Compccmp 21994
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-fin 8509  df-top 21502  df-cld 21627  df-ntr 21628  df-cls 21629  df-lp 21744  df-cmp 21995
This theorem is referenced by:  poimirlem30  35032  fourierdlem42  42717
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