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Mirrors > Home > NFE Home > Th. List > cnvkexg | Unicode version |
Description: The Kuratowski converse of a set is a set. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
cnvkexg | k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvkeq 4215 | . . 3 k k | |
2 | 1 | eleq1d 2419 | . 2 k k |
3 | ax-cnv 4080 | . . 3 | |
4 | inss1 3475 | . . . . . . . 8 k k | |
5 | cnvkssvvk 4275 | . . . . . . . 8 k k | |
6 | eqrelk 4212 | . . . . . . . 8 k k k k k k k k | |
7 | 4, 5, 6 | mp2an 653 | . . . . . . 7 k k k k |
8 | vex 2862 | . . . . . . . . . . 11 | |
9 | vex 2862 | . . . . . . . . . . 11 | |
10 | 8, 9 | opkelxpk 4248 | . . . . . . . . . . 11 k |
11 | 8, 9, 10 | mpbir2an 886 | . . . . . . . . . 10 k |
12 | elin 3219 | . . . . . . . . . 10 k k | |
13 | 11, 12 | mpbiran 884 | . . . . . . . . 9 k |
14 | 8, 9 | opkelcnvk 4250 | . . . . . . . . 9 k |
15 | 13, 14 | bibi12i 306 | . . . . . . . 8 k k |
16 | 15 | 2albii 1567 | . . . . . . 7 k k |
17 | 7, 16 | bitri 240 | . . . . . 6 k k |
18 | 17 | biimpri 197 | . . . . 5 k k |
19 | vvex 4109 | . . . . . . 7 | |
20 | xpkvexg 4285 | . . . . . . 7 k | |
21 | 19, 20 | ax-mp 8 | . . . . . 6 k |
22 | vex 2862 | . . . . . 6 | |
23 | 21, 22 | inex 4105 | . . . . 5 k |
24 | 18, 23 | syl6eqelr 2442 | . . . 4 k |
25 | 24 | exlimiv 1634 | . . 3 k |
26 | 3, 25 | ax-mp 8 | . 2 k |
27 | 2, 26 | vtoclg 2914 | 1 k |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wal 1540 wex 1541 wceq 1642 wcel 1710 cvv 2859 cin 3208 wss 3257 copk 4057 k cxpk 4174 kccnvk 4175 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 df-cnvk 4186 |
This theorem is referenced by: cnvkex 4287 xpkexg 4288 cokexg 4309 imagekexg 4311 |
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