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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distop 13624 |
. . . . 5
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2 | unipw 4219 |
. . . . . . 7
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3 | 2 | eqcomi 2181 |
. . . . . 6
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4 | 3 | iscld 13642 |
. . . . 5
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5 | 1, 4 | syl 14 |
. . . 4
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6 | difss 3263 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | elpw2g 4158 |
. . . . . 6
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8 | 6, 7 | mpbiri 168 |
. . . . 5
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9 | 8 | biantrud 304 |
. . . 4
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10 | 5, 9 | bitr4d 191 |
. . 3
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11 | velpw 3584 |
. . 3
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12 | 10, 11 | bitr4di 198 |
. 2
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13 | 12 | eqrdv 2175 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-top 13537 df-cld 13634 |
This theorem is referenced by: sn0cld 13676 |
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