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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 2735 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | cldss 23053 | . . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽) |
3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ 𝐽) |
4 | dfss4 4275 | . . . . 5 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥) | |
5 | 3, 4 | sylib 218 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥) |
6 | 1 | topopn 22928 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
7 | 1 | difopn 23058 | . . . . . 6 ⊢ ((∪ 𝐽 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) |
8 | 6, 7 | sylan 580 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) |
9 | id 22 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top) | |
10 | 9 | sgsiga 34123 | . . . . . . 7 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
12 | elex 3499 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V) | |
13 | sigagensiga 34122 | . . . . . . . 8 ⊢ (𝐽 ∈ V → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) | |
14 | baselsiga 34096 | . . . . . . . 8 ⊢ ((sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽)) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (sigaGen‘𝐽)) |
16 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → ∪ 𝐽 ∈ (sigaGen‘𝐽)) |
17 | elsigagen 34128 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) | |
18 | difelsiga 34114 | . . . . . 6 ⊢ (((sigaGen‘𝐽) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐽 ∈ (sigaGen‘𝐽) ∧ (∪ 𝐽 ∖ 𝑥) ∈ (sigaGen‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽)) | |
19 | 11, 16, 17, 18 | syl3anc 1370 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽)) |
20 | 8, 19 | syldan 591 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ∈ (sigaGen‘𝐽)) |
21 | 5, 20 | eqeltrrd 2840 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (sigaGen‘𝐽)) |
22 | 21 | ex 412 | . 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ (sigaGen‘𝐽))) |
23 | 22 | ssrdv 4001 | 1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 ∪ cuni 4912 ran crn 5690 ‘cfv 6563 Topctop 22915 Clsdccld 23040 sigAlgebracsiga 34089 sigaGencsigagen 34119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 df-top 22916 df-cld 23043 df-siga 34090 df-sigagen 34120 |
This theorem is referenced by: sxbrsigalem4 34269 sibfinima 34321 sibfof 34322 orvccel 34444 |
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