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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 20640 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld 20641 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
4 | 3 | simprbda 651 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋) |
5 | 1, 4 | mpancom 700 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 Clsdccld 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-top 20521 df-cld 20633 |
This theorem is referenced by: cldss2 20644 iincld 20653 uncld 20655 cldcls 20656 iuncld 20659 clsval2 20664 clsss3 20673 clsss2 20686 opncldf1 20698 restcldr 20788 lmcld 20917 nrmsep2 20970 nrmsep 20971 isnrm2 20972 regsep2 20990 cmpcld 21015 dfcon2 21032 concompclo 21048 cldllycmp 21108 txcld 21216 ptcld 21226 imasncld 21304 kqcldsat 21346 kqnrmlem1 21356 kqnrmlem2 21357 nrmhmph 21407 ufildr 21545 metnrmlem1a 22469 metnrmlem1 22470 metnrmlem2 22471 metnrmlem3 22472 cnheiborlem 22561 cmetss 22921 bcthlem5 22933 cldssbrsiga 29577 clsun 31493 cldregopn 31496 mblfinlem3 32618 mblfinlem4 32619 ismblfin 32620 cmpfiiin 36278 kelac1 36651 stoweidlem18 38911 stoweidlem57 38950 |
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