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Theorem inxpk 4277
 Description: The intersection of two Kuratowski cross products. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
inxpk k k k

Proof of Theorem inxpk
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3475 . . 3 k k k
2 xpkssvvk 4210 . . 3 k k
31, 2sstri 3281 . 2 k k k
4 xpkssvvk 4210 . 2 k k
5 an4 797 . . 3
6 elin 3219 . . . 4 k k k k
7 vex 2862 . . . . . 6
8 vex 2862 . . . . . 6
97, 8opkelxpk 4248 . . . . 5 k
107, 8opkelxpk 4248 . . . . 5 k
119, 10anbi12i 678 . . . 4 k k
126, 11bitri 240 . . 3 k k
137, 8opkelxpk 4248 . . . 4 k
14 elin 3219 . . . . 5
15 elin 3219 . . . . 5
1614, 15anbi12i 678 . . . 4
1713, 16bitri 240 . . 3 k
185, 12, 173bitr4i 268 . 2 k k k
193, 4, 18eqrelkriiv 4213 1 k k k
 Colors of variables: wff setvar class Syntax hints:   wa 358   wceq 1642   wcel 1710  cvv 2859   cin 3208  copk 4057   k cxpk 4174 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-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-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:  xpkexg  4288
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