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Mirrors > Home > MPE Home > Th. List > iscld | Structured version Visualization version Unicode version |
Description: The predicate " is a closed set." (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 |
Ref | Expression |
---|---|
iscld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 | |
2 | 1 | cldval 20827 | . . . 4 |
3 | 2 | eleq2d 2687 | . . 3 |
4 | difeq2 3722 | . . . . 5 | |
5 | 4 | eleq1d 2686 | . . . 4 |
6 | 5 | elrab 3363 | . . 3 |
7 | 3, 6 | syl6bb 276 | . 2 |
8 | 1 | topopn 20711 | . . . 4 |
9 | elpw2g 4827 | . . . 4 | |
10 | 8, 9 | syl 17 | . . 3 |
11 | 10 | anbi1d 741 | . 2 |
12 | 7, 11 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 crab 2916 cdif 3571 wss 3574 cpw 4158 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: iscld2 20832 cldss 20833 cldopn 20835 topcld 20839 discld 20893 indiscld 20895 restcld 20976 ordtcld1 21001 ordtcld2 21002 hauscmp 21210 txcld 21406 ptcld 21416 qtopcld 21516 opnsubg 21911 sszcld 22620 stoweidlem57 40274 |
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