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Theorem ssetkex 4294
 Description: The Kuratowski subset relationship is a set. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
ssetkex Sk

Proof of Theorem ssetkex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-sset 4082 . 2
2 inss1 3475 . . . . . . 7 k k
3 ssetkssvvk 4278 . . . . . . 7 Sk k
4 eqrelk 4212 . . . . . . 7 k k Sk k k Sk k Sk
52, 3, 4mp2an 653 . . . . . 6 k Sk k Sk
6 vex 2862 . . . . . . . . . 10
7 vex 2862 . . . . . . . . . 10
86, 7opkelxpk 4248 . . . . . . . . . 10 k
96, 7, 8mpbir2an 886 . . . . . . . . 9 k
10 elin 3219 . . . . . . . . 9 k k
119, 10mpbiran 884 . . . . . . . 8 k
12 opkelssetkg 4268 . . . . . . . . . 10 Sk
136, 7, 12mp2an 653 . . . . . . . . 9 Sk
14 dfss2 3262 . . . . . . . . 9
1513, 14bitri 240 . . . . . . . 8 Sk
1611, 15bibi12i 306 . . . . . . 7 k Sk
17162albii 1567 . . . . . 6 k Sk
185, 17bitri 240 . . . . 5 k Sk
1918biimpri 197 . . . 4 k Sk
20 vvex 4109 . . . . . 6
21 xpkvexg 4285 . . . . . 6 k
2220, 21ax-mp 8 . . . . 5 k
23 vex 2862 . . . . 5
2422, 23inex 4105 . . . 4 k
2519, 24syl6eqelr 2442 . . 3 Sk
2625exlimiv 1634 . 2 Sk
271, 26ax-mp 8 1 Sk
 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   Sk cssetk 4183 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-sset 4082  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-ssetk 4193 This theorem is referenced by:  imagekexg  4311  idkex  4314  uniexg  4316  intexg  4319  setswithex  4322  pwexg  4328  addcexlem  4382  nncex  4396  nnsucelrlem1  4424  nndisjeq  4429  ltfinex  4464  ssfin  4470  ncfinraiselem2  4480  ncfinlowerlem1  4482  tfinrelkex  4487  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelkex  4525  tfinnnlem1  4533  spfinex  4537  opexg  4587  proj2exg  4592  setconslem5  4735  1stex  4739  swapex  4742  ssetex  4744
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