Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unicls | Structured version Visualization version GIF version |
Description: The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
Ref | Expression |
---|---|
unicls.1 | ⊢ 𝐽 ∈ Top |
unicls.2 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
unicls | ⊢ ∪ (Clsd‘𝐽) = 𝑋 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unicls.2 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | cldss2 21566 | . . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
3 | sspwuni 5013 | . . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋) | |
4 | 2, 3 | mpbi 231 | . 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋 |
5 | unicls.1 | . . 3 ⊢ 𝐽 ∈ Top | |
6 | 1 | topcld 21571 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ 𝑋 ∈ (Clsd‘𝐽) |
8 | unissel 4860 | . 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋) | |
9 | 4, 7, 8 | mp2an 688 | 1 ⊢ ∪ (Clsd‘𝐽) = 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ‘cfv 6348 Topctop 21429 Clsdccld 21552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-top 21430 df-cld 21555 |
This theorem is referenced by: sxbrsigalem3 31429 sxbrsigalem4 31444 |
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