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Theorem cnvkexg 4286
Description: The Kuratowski converse of a set is a set. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
cnvkexg k

Proof of Theorem cnvkexg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvkeq 4215 . . 3 k k
21eleq1d 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
74, 5, 6mp2an 653 . . . . . . 7 k k k k
8 vex 2862 . . . . . . . . . . 11
9 vex 2862 . . . . . . . . . . 11
108, 9opkelxpk 4248 . . . . . . . . . . 11 k
118, 9, 10mpbir2an 886 . . . . . . . . . 10 k
12 elin 3219 . . . . . . . . . 10 k k
1311, 12mpbiran 884 . . . . . . . . 9 k
148, 9opkelcnvk 4250 . . . . . . . . 9 k
1513, 14bibi12i 306 . . . . . . . 8 k k
16152albii 1567 . . . . . . 7 k k
177, 16bitri 240 . . . . . 6 k k
1817biimpri 197 . . . . 5 k k
19 vvex 4109 . . . . . . 7
20 xpkvexg 4285 . . . . . . 7 k
2119, 20ax-mp 5 . . . . . 6 k
22 vex 2862 . . . . . 6
2321, 22inex 4105 . . . . 5 k
2418, 23syl6eqelr 2442 . . . 4 k
2524exlimiv 1634 . . 3 k
263, 25ax-mp 5 . 2 k
272, 26vtoclg 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-mp 5  ax-1 6  ax-2 7  ax-3 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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