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Theorem elssetkg 4269
 Description: Membership via the Kuratowski subset relationship. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
elssetkg Sk

Proof of Theorem elssetkg
StepHypRef Expression
1 snex 4111 . . 3
2 opkelssetkg 4268 . . 3 Sk
31, 2mpan 651 . 2 Sk
4 snssg 3844 . . 3
54bicomd 192 . 2
63, 5sylan9bbr 681 1 Sk
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wcel 1710  cvv 2859   wss 3257  csn 3737  copk 4057   Sk cssetk 4183 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-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-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-ssetk 4193 This theorem is referenced by:  elssetk  4270  opkelimagekg  4271  setswith  4321
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