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Mirrors > Home > MPE Home > Th. List > clscld | Structured version Visualization version Unicode version |
Description: The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
clscld.1 |
Ref | Expression |
---|---|
clscld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | . . 3 | |
2 | 1 | clsval 20841 | . 2 |
3 | 1 | topcld 20839 | . . . . . 6 |
4 | 3 | anim1i 592 | . . . . 5 |
5 | sseq2 3627 | . . . . . 6 | |
6 | 5 | elrab 3363 | . . . . 5 |
7 | 4, 6 | sylibr 224 | . . . 4 |
8 | ne0i 3921 | . . . 4 | |
9 | 7, 8 | syl 17 | . . 3 |
10 | ssrab2 3687 | . . 3 | |
11 | intcld 20844 | . . 3 | |
12 | 9, 10, 11 | sylancl 694 | . 2 |
13 | 2, 12 | eqeltrd 2701 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 crab 2916 wss 3574 c0 3915 cuni 4436 cint 4475 cfv 5888 ctop 20698 ccld 20820 ccl 20822 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-int 4476 df-iun 4522 df-iin 4523 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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-top 20699 df-cld 20823 df-cls 20825 |
This theorem is referenced by: clsf 20852 clsss3 20863 iscld3 20868 clsidm 20871 restcls 20985 cncls2i 21074 nrmsep 21161 lpcls 21168 regsep2 21180 hauscmplem 21209 hausllycmp 21297 txcls 21407 ptclsg 21418 regr1lem 21542 kqreglem1 21544 kqreglem2 21545 kqnrmlem1 21546 kqnrmlem2 21547 fclscmpi 21833 tgptsmscld 21954 cnllycmp 22755 clsocv 23049 cmpcmet 23116 cncmet 23119 limcnlp 23642 clsun 32323 cldregopn 32326 heibor1lem 33608 |
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