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Theorem bwth 22777
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 = βˆͺ 𝐽
Assertion
Ref Expression
bwth ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) β†’ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐽   π‘₯,𝑋

Proof of Theorem bwth
Dummy variables π‘œ 𝑏 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm3.24 404 . . . . . . 7 Β¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
21a1i 11 . . . . . 6 (𝑏 ∈ 𝑧 β†’ Β¬ ((𝐴 ∩ 𝑏) ∈ Fin ∧ Β¬ (𝐴 ∩ 𝑏) ∈ Fin))
32nrex 3078 . . . . 5 Β¬ βˆƒπ‘ ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4 r19.29 3118 . . . . 5 ((βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin) β†’ βˆƒπ‘ ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ Β¬ (𝐴 ∩ 𝑏) ∈ Fin))
53, 4mto 196 . . . 4 Β¬ (βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
65a1i 11 . . 3 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) β†’ Β¬ (βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin))
76nrex 3078 . 2 Β¬ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
8 ralnex 3076 . . . . . 6 (βˆ€π‘₯ ∈ 𝑋 Β¬ π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ Β¬ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄))
9 cmptop 22762 . . . . . . 7 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
10 bwt2.1 . . . . . . . . . . 11 𝑋 = βˆͺ 𝐽
1110islp3 22513 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…)))
12113expa 1119 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…)))
1312notbid 318 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) ∧ π‘₯ ∈ 𝑋) β†’ (Β¬ π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…)))
1413ralbidva 3173 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 Β¬ π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…)))
159, 14sylan 581 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 Β¬ π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…)))
168, 15bitr3id 285 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋) β†’ (Β¬ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…)))
17 rexanali 3106 . . . . . . . . 9 (βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ Β¬ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) ↔ Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…))
18 nne 2948 . . . . . . . . . . . 12 (Β¬ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ… ↔ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) = βˆ…)
19 vex 3452 . . . . . . . . . . . . 13 π‘₯ ∈ V
20 sneq 4601 . . . . . . . . . . . . . . . 16 (π‘œ = π‘₯ β†’ {π‘œ} = {π‘₯})
2120difeq2d 4087 . . . . . . . . . . . . . . 15 (π‘œ = π‘₯ β†’ (𝐴 βˆ– {π‘œ}) = (𝐴 βˆ– {π‘₯}))
2221ineq2d 4177 . . . . . . . . . . . . . 14 (π‘œ = π‘₯ β†’ (𝑏 ∩ (𝐴 βˆ– {π‘œ})) = (𝑏 ∩ (𝐴 βˆ– {π‘₯})))
2322eqeq1d 2739 . . . . . . . . . . . . 13 (π‘œ = π‘₯ β†’ ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… ↔ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) = βˆ…))
2419, 23spcev 3568 . . . . . . . . . . . 12 ((𝑏 ∩ (𝐴 βˆ– {π‘₯})) = βˆ… β†’ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)
2518, 24sylbi 216 . . . . . . . . . . 11 (Β¬ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ… β†’ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)
2625anim2i 618 . . . . . . . . . 10 ((π‘₯ ∈ 𝑏 ∧ Β¬ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) β†’ (π‘₯ ∈ 𝑏 ∧ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…))
2726reximi 3088 . . . . . . . . 9 (βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ Β¬ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) β†’ βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…))
2817, 27sylbir 234 . . . . . . . 8 (Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) β†’ βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…))
2928ralimi 3087 . . . . . . 7 (βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…))
3010cmpcov2 22757 . . . . . . . 8 ((𝐽 ∈ Comp ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…))
3130ex 414 . . . . . . 7 (𝐽 ∈ Comp β†’ (βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 ∧ βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)))
3229, 31syl5 34 . . . . . 6 (𝐽 ∈ Comp β†’ (βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)))
3332adantr 482 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘ ∈ 𝐽 (π‘₯ ∈ 𝑏 β†’ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ…) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)))
3416, 33sylbid 239 . . . 4 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋) β†’ (Β¬ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)))
35343adant3 1133 . . 3 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) β†’ (Β¬ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…)))
36 elinel2 4161 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) β†’ 𝑧 ∈ Fin)
37 sseq2 3975 . . . . . . . . . . . 12 (𝑋 = βˆͺ 𝑧 β†’ (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝑧))
3837biimpac 480 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ 𝑋 = βˆͺ 𝑧) β†’ 𝐴 βŠ† βˆͺ 𝑧)
39 infssuni 9294 . . . . . . . . . . . . 13 ((Β¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin ∧ 𝐴 βŠ† βˆͺ 𝑧) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
40393expa 1119 . . . . . . . . . . . 12 (((Β¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin) ∧ 𝐴 βŠ† βˆͺ 𝑧) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4140ancoms 460 . . . . . . . . . . 11 ((𝐴 βŠ† βˆͺ 𝑧 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4238, 41sylan 581 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝑋 = βˆͺ 𝑧) ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑧 ∈ Fin)) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4342an42s 660 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) ∧ (𝑧 ∈ Fin ∧ 𝑋 = βˆͺ 𝑧)) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4443anassrs 469 . . . . . . . 8 ((((𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ Fin) ∧ 𝑋 = βˆͺ 𝑧) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4536, 44sylanl2 680 . . . . . . 7 ((((𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = βˆͺ 𝑧) β†’ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
46 0fin 9122 . . . . . . . . . . . 12 βˆ… ∈ Fin
47 eleq1 2826 . . . . . . . . . . . 12 ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… β†’ ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) ∈ Fin ↔ βˆ… ∈ Fin))
4846, 47mpbiri 258 . . . . . . . . . . 11 ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… β†’ (𝑏 ∩ (𝐴 βˆ– {π‘œ})) ∈ Fin)
49 snfi 8995 . . . . . . . . . . 11 {π‘œ} ∈ Fin
50 unfi 9123 . . . . . . . . . . 11 (((𝑏 ∩ (𝐴 βˆ– {π‘œ})) ∈ Fin ∧ {π‘œ} ∈ Fin) β†’ ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ}) ∈ Fin)
5148, 49, 50sylancl 587 . . . . . . . . . 10 ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… β†’ ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ}) ∈ Fin)
52 ssun1 4137 . . . . . . . . . . . 12 𝑏 βŠ† (𝑏 βˆͺ {π‘œ})
53 ssun1 4137 . . . . . . . . . . . . 13 𝐴 βŠ† (𝐴 βˆͺ {π‘œ})
54 undif1 4440 . . . . . . . . . . . . 13 ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ}) = (𝐴 βˆͺ {π‘œ})
5553, 54sseqtrri 3986 . . . . . . . . . . . 12 𝐴 βŠ† ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ})
56 ss2in 4201 . . . . . . . . . . . 12 ((𝑏 βŠ† (𝑏 βˆͺ {π‘œ}) ∧ 𝐴 βŠ† ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ})) β†’ (𝑏 ∩ 𝐴) βŠ† ((𝑏 βˆͺ {π‘œ}) ∩ ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ})))
5752, 55, 56mp2an 691 . . . . . . . . . . 11 (𝑏 ∩ 𝐴) βŠ† ((𝑏 βˆͺ {π‘œ}) ∩ ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ}))
58 incom 4166 . . . . . . . . . . 11 (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59 undir 4241 . . . . . . . . . . 11 ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ}) = ((𝑏 βˆͺ {π‘œ}) ∩ ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ}))
6057, 58, 593sstr4i 3992 . . . . . . . . . 10 (𝐴 ∩ 𝑏) βŠ† ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ})
61 ssfi 9124 . . . . . . . . . 10 ((((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ}) ∈ Fin ∧ (𝐴 ∩ 𝑏) βŠ† ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ})) β†’ (𝐴 ∩ 𝑏) ∈ Fin)
6251, 60, 61sylancl 587 . . . . . . . . 9 ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… β†’ (𝐴 ∩ 𝑏) ∈ Fin)
6362exlimiv 1934 . . . . . . . 8 (βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… β†’ (𝐴 ∩ 𝑏) ∈ Fin)
6463ralimi 3087 . . . . . . 7 (βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… β†’ βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin)
6545, 64anim12ci 615 . . . . . 6 (((((𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) ∧ 𝑋 = βˆͺ 𝑧) ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…) β†’ (βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin))
6665expl 459 . . . . 5 (((𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) ∧ 𝑧 ∈ (𝒫 𝐽 ∩ Fin)) β†’ ((𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…) β†’ (βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)))
6766reximdva 3166 . . . 4 ((𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) β†’ (βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)))
68673adant1 1131 . . 3 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) β†’ (βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = βˆͺ 𝑧 ∧ βˆ€π‘ ∈ 𝑧 βˆƒπ‘œ(𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ…) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)))
6935, 68syld 47 . 2 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) β†’ (Β¬ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)))
707, 69mt3i 149 1 ((𝐽 ∈ Comp ∧ 𝐴 βŠ† 𝑋 ∧ Β¬ 𝐴 ∈ Fin) β†’ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074   βˆ– cdif 3912   βˆͺ cun 3913   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  {csn 4591  βˆͺ cuni 4870  β€˜cfv 6501  Fincfn 8890  Topctop 22258  limPtclp 22501  Compccmp 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-fin 8894  df-top 22259  df-cld 22386  df-ntr 22387  df-cls 22388  df-lp 22503  df-cmp 22754
This theorem is referenced by:  poimirlem30  36137  fourierdlem42  44464
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