NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  elxpk Unicode version

Theorem elxpk 4196
Description: Membership in a Kuratowski cross product. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
elxpk k
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem elxpk
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . 2 k
2 opkex 4113 . . . . 5
3 eleq1 2413 . . . . 5
42, 3mpbiri 224 . . . 4
54adantr 451 . . 3
65exlimivv 1635 . 2
7 eqeq1 2359 . . . . 5
87anbi1d 685 . . . 4
982exbidv 1628 . . 3
10 df-xpk 4185 . . 3 k
119, 10elab2g 2987 . 2 k
121, 6, 11pm5.21nii 342 1 k
Colors of variables: wff setvar class
Syntax hints:   wb 176   wa 358  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-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-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:  elxpk2  4197  elvvk  4207  xpkvexg  4285  sikexlem  4295  insklem  4304
  Copyright terms: Public domain W3C validator