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Theorem opkabssvvk 4208
 Description: Any Kuratowski ordered pair abstraction is a subset of k . (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
opkabssvvk k
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,,)

Proof of Theorem opkabssvvk
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2353 . . . . . . 7
2 vex 2862 . . . . . . . 8
3 vex 2862 . . . . . . . 8
4 opkeq12 4061 . . . . . . . . 9
54eqeq2d 2364 . . . . . . . 8
62, 3, 5spc2ev 2947 . . . . . . 7
71, 6ax-mp 5 . . . . . 6
8 elvvk 4207 . . . . . 6 k
97, 8mpbir 200 . . . . 5 k
10 eleq1 2413 . . . . 5 k k
119, 10mpbiri 224 . . . 4 k
1211adantr 451 . . 3 k
1312exlimivv 1635 . 2 k
1413abssi 3341 1 k
 Colors of variables: wff setvar class Syntax hints:   wa 358  wex 1541   wceq 1642   wcel 1710  cab 2339  cvv 2859   wss 3257  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:  opkabssvvki  4209
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