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Mirrors > Home > MPE Home > Th. List > opncld | Structured version Visualization version Unicode version |
Description: The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.) |
Ref | Expression |
---|---|
iscld.1 |
Ref | Expression |
---|---|
opncld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . 2 | |
2 | iscld.1 | . . . 4 | |
3 | 2 | eltopss 20712 | . . 3 |
4 | 2 | isopn2 20836 | . . 3 |
5 | 3, 4 | syldan 487 | . 2 |
6 | 1, 5 | mpbid 222 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cdif 3571 wss 3574 cuni 4436 cfv 5888 ctop 20698 ccld 20820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-top 20699 df-cld 20823 |
This theorem is referenced by: iincld 20843 iuncld 20849 clsval2 20854 cmntrcld 20867 elcls 20877 opncldf1 20888 opncldf2 20889 restcld 20976 iscncl 21073 pnrmopn 21147 isnrm2 21162 isnrm3 21163 isreg2 21181 hauscmplem 21209 conndisj 21219 hausllycmp 21297 1stckgen 21357 txkgen 21455 qtoprest 21520 qtopcmap 21522 icopnfcld 22571 lebnumlem1 22760 bcth3 23128 sxbrsigalem3 30334 pconnconn 31213 cvmscld 31255 cldbnd 32321 mblfinlem3 33448 mblfinlem4 33449 |
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