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Mirrors > Home > ILE Home > Th. List > discld | Unicode version |
Description: The open sets of a discrete topology are closed and its closed sets are open. (Contributed by FL, 7-Jun-2007.) (Revised by Mario Carneiro, 7-Apr-2015.) |
Ref | Expression |
---|---|
discld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distop 12181 | . . . . 5 | |
2 | unipw 4109 | . . . . . . 7 | |
3 | 2 | eqcomi 2121 | . . . . . 6 |
4 | 3 | iscld 12199 | . . . . 5 |
5 | 1, 4 | syl 14 | . . . 4 |
6 | difss 3172 | . . . . . 6 | |
7 | elpw2g 4051 | . . . . . 6 | |
8 | 6, 7 | mpbiri 167 | . . . . 5 |
9 | 8 | biantrud 302 | . . . 4 |
10 | 5, 9 | bitr4d 190 | . . 3 |
11 | velpw 3487 | . . 3 | |
12 | 10, 11 | syl6bbr 197 | . 2 |
13 | 12 | eqrdv 2115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cdif 3038 wss 3041 cpw 3480 cuni 3706 cfv 5093 ctop 12091 ccld 12188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-top 12092 df-cld 12191 |
This theorem is referenced by: sn0cld 12233 |
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