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Mirrors > Home > MPE Home > Th. List > Mathboxes > cldssbrsiga | Structured version Visualization version GIF version |
Description: A Borel Algebra contains all closed sets of its base topology. (Contributed by Thierry Arnoux, 27-Mar-2017.) |
Ref | Expression |
---|---|
cldssbrsiga | ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | cldss 21634 | . . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽) |
3 | 2 | adantl 485 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ 𝐽) |
4 | dfss4 4185 | . . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥) | |
5 | 3, 4 | sylib 221 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥) |
6 | 1 | topopn 21511 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
7 | 1 | difopn 21639 | . . . . . 6 ⊢ ((∪ 𝐽 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) |
8 | 6, 7 | sylan 583 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) |
9 | id 22 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top) | |
10 | 9 | sgsiga 31511 | . . . . . . 7 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
11 | 10 | adantr 484 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
12 | elex 3459 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
13 | sigagensiga 31510 | . . . . . . . 8 ⊢ (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) | |
14 | baselsiga 31484 | . . . . . . . 8 ⊢ ((sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽)) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (sigaGen‘𝐽)) |
16 | 15 | adantr 484 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽)) |
17 | elsigagen 31516 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) | |
18 | difelsiga 31502 | . . . . . 6 ⊢ (((sigaGen‘𝐽) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐽 ∈ (sigaGen‘𝐽) ∧ (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽)) | |
19 | 11, 16, 17, 18 | syl3anc 1368 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽)) |
20 | 8, 19 | syldan 594 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽)) |
21 | 5, 20 | eqeltrrd 2891 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽)) |
22 | 21 | ex 416 | . 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽))) |
23 | 22 | ssrdv 3921 | 1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 Topctop 21498 Clsdccld 21621 sigAlgebracsiga 31477 sigaGencsigagen 31507 |
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 ax-inf2 9088 ax-ac2 9874 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 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-rmo 3114 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-se 5479 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-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-top 21499 df-cld 21624 df-siga 31478 df-sigagen 31508 |
This theorem is referenced by: sxbrsigalem4 31655 sibfinima 31707 sibfof 31708 orvccel 31830 |
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