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Theorem xpkvexg 4285
 Description: The Kuratowski cross product of with a set is a set. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
xpkvexg k

Proof of Theorem xpkvexg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpkeq2 4199 . . 3 k k
21eleq1d 2419 . 2 k k
3 ax-xp 4079 . . 3
4 isset 2863 . . . 4 k k
5 dfcleq 2347 . . . . . 6 k k
6 elxpk 4196 . . . . . . . . 9 k
7 vex 2862 . . . . . . . . . . . 12
87biantrur 492 . . . . . . . . . . 11
98anbi2i 675 . . . . . . . . . 10
1092exbii 1583 . . . . . . . . 9
116, 10bitr4i 243 . . . . . . . 8 k
1211bibi2i 304 . . . . . . 7 k
1312albii 1566 . . . . . 6 k
145, 13bitri 240 . . . . 5 k
1514exbii 1582 . . . 4 k
164, 15bitri 240 . . 3 k
173, 16mpbir 200 . 2 k
182, 17vtoclg 2914 1 k
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  cvv 2859  copk 4057   k cxpk 4174 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-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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 This theorem is referenced by:  cnvkexg  4286  xpkexg  4288  ssetkex  4294
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