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| Mirrors > Home > MPE Home > Th. List > cldss | Structured version Visualization version GIF version | ||
| Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
| Ref | Expression |
|---|---|
| iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| cldss | ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cldrcl 23034 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
| 2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
| 3 | 2 | iscld 23035 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
| 4 | 3 | simprbda 498 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋) |
| 5 | 1, 4 | mpancom 688 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 Clsdccld 23024 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-top 22900 df-cld 23027 |
| This theorem is referenced by: cldss2 23038 iincld 23047 uncld 23049 cldcls 23050 iuncld 23053 clsval2 23058 clsss3 23067 clsss2 23080 opncldf1 23092 restcldr 23182 lmcld 23311 nrmsep2 23364 nrmsep 23365 isnrm2 23366 regsep2 23384 cmpcld 23410 dfconn2 23427 conncompclo 23443 cldllycmp 23503 txcld 23611 ptcld 23621 imasncld 23699 kqcldsat 23741 kqnrmlem1 23751 kqnrmlem2 23752 nrmhmph 23802 ufildr 23939 metnrmlem1a 24880 metnrmlem1 24881 metnrmlem2 24882 metnrmlem3 24883 cnheiborlem 24986 cmetss 25350 bcthlem5 25362 cldssbrsiga 34188 clsun 36329 cldregopn 36332 pibt2 37418 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 cmpfiiin 42708 kelac1 43075 stoweidlem18 46033 stoweidlem57 46072 restcls2lem 48810 |
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