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Mirrors > Home > MPE Home > Th. List > topcld | Structured version Visualization version Unicode version |
Description: The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.) |
Ref | Expression |
---|---|
iscld.1 |
Ref | Expression |
---|---|
topcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difid 3948 | . . . 4 | |
2 | 0opn 20709 | . . . 4 | |
3 | 1, 2 | syl5eqel 2705 | . . 3 |
4 | ssid 3624 | . . 3 | |
5 | 3, 4 | jctil 560 | . 2 |
6 | iscld.1 | . . 3 | |
7 | 6 | iscld 20831 | . 2 |
8 | 5, 7 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cdif 3571 wss 3574 c0 3915 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: clsval 20841 riincld 20848 clscld 20851 clstop 20873 cldmre 20882 indiscld 20895 isconn2 21217 cnmpt2pc 22727 rlmbn 23157 ubthlem1 27726 unicls 29949 cmpfiiin 37260 kelac1 37633 |
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