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 21640 | . . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
3 | sspwuni 5024 | . . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋) | |
4 | 2, 3 | mpbi 232 | . 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋 |
5 | unicls.1 | . . 3 ⊢ 𝐽 ∈ Top | |
6 | 1 | topcld 21645 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ 𝑋 ∈ (Clsd‘𝐽) |
8 | unissel 4871 | . 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋) | |
9 | 4, 7, 8 | mp2an 690 | 1 ⊢ ∪ (Clsd‘𝐽) = 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 𝒫 cpw 4541 ∪ cuni 4840 ‘cfv 6357 Topctop 21503 Clsdccld 21626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-top 21504 df-cld 21629 |
This theorem is referenced by: sxbrsigalem3 31532 sxbrsigalem4 31547 |
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