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Theorem bwth 22914
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 3075 . . . . 5 Β¬ βˆƒπ‘ ∈ 𝑧 ((𝐴 ∩ 𝑏) ∈ Fin ∧ Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
4 r19.29 3115 . . . . 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 3075 . 2 Β¬ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)(βˆ€π‘ ∈ 𝑧 (𝐴 ∩ 𝑏) ∈ Fin ∧ βˆƒπ‘ ∈ 𝑧 Β¬ (𝐴 ∩ 𝑏) ∈ Fin)
8 ralnex 3073 . . . . . 6 (βˆ€π‘₯ ∈ 𝑋 Β¬ π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄) ↔ Β¬ βˆƒπ‘₯ ∈ 𝑋 π‘₯ ∈ ((limPtβ€˜π½)β€˜π΄))
9 cmptop 22899 . . . . . . 7 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
10 bwt2.1 . . . . . . . . . . 11 𝑋 = βˆͺ 𝐽
1110islp3 22650 . . . . . . . . . 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 3176 . . . . . . 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 3103 . . . . . . . . 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 2945 . . . . . . . . . . . 12 (Β¬ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) β‰  βˆ… ↔ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) = βˆ…)
19 vex 3479 . . . . . . . . . . . . 13 π‘₯ ∈ V
20 sneq 4639 . . . . . . . . . . . . . . . 16 (π‘œ = π‘₯ β†’ {π‘œ} = {π‘₯})
2120difeq2d 4123 . . . . . . . . . . . . . . 15 (π‘œ = π‘₯ β†’ (𝐴 βˆ– {π‘œ}) = (𝐴 βˆ– {π‘₯}))
2221ineq2d 4213 . . . . . . . . . . . . . 14 (π‘œ = π‘₯ β†’ (𝑏 ∩ (𝐴 βˆ– {π‘œ})) = (𝑏 ∩ (𝐴 βˆ– {π‘₯})))
2322eqeq1d 2735 . . . . . . . . . . . . 13 (π‘œ = π‘₯ β†’ ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) = βˆ… ↔ (𝑏 ∩ (𝐴 βˆ– {π‘₯})) = βˆ…))
2419, 23spcev 3597 . . . . . . . . . . . 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 3085 . . . . . . . . 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 3084 . . . . . . 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 22894 . . . . . . . 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 4197 . . . . . . . 8 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) β†’ 𝑧 ∈ Fin)
37 sseq2 4009 . . . . . . . . . . . 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 9343 . . . . . . . . . . . . 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 9171 . . . . . . . . . . . 12 βˆ… ∈ Fin
47 eleq1 2822 . . . . . . . . . . . 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 9044 . . . . . . . . . . 11 {π‘œ} ∈ Fin
50 unfi 9172 . . . . . . . . . . 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 4173 . . . . . . . . . . . 12 𝑏 βŠ† (𝑏 βˆͺ {π‘œ})
53 ssun1 4173 . . . . . . . . . . . . 13 𝐴 βŠ† (𝐴 βˆͺ {π‘œ})
54 undif1 4476 . . . . . . . . . . . . 13 ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ}) = (𝐴 βˆͺ {π‘œ})
5553, 54sseqtrri 4020 . . . . . . . . . . . 12 𝐴 βŠ† ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ})
56 ss2in 4237 . . . . . . . . . . . 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 4202 . . . . . . . . . . 11 (𝐴 ∩ 𝑏) = (𝑏 ∩ 𝐴)
59 undir 4277 . . . . . . . . . . 11 ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ}) = ((𝑏 βˆͺ {π‘œ}) ∩ ((𝐴 βˆ– {π‘œ}) βˆͺ {π‘œ}))
6057, 58, 593sstr4i 4026 . . . . . . . . . 10 (𝐴 ∩ 𝑏) βŠ† ((𝑏 ∩ (𝐴 βˆ– {π‘œ})) βˆͺ {π‘œ})
61 ssfi 9173 . . . . . . . . . 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 3084 . . . . . . 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 3169 . . . 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 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  Fincfn 8939  Topctop 22395  limPtclp 22638  Compccmp 22890
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525  df-lp 22640  df-cmp 22891
This theorem is referenced by:  poimirlem30  36518  fourierdlem42  44865
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