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

Theorem elssetk 4270
 Description: Membership via the Kuratowski subset relationship. (Contributed by SF, 13-Jan-2015.)
Hypotheses
Ref Expression
elssetk.1 A V
elssetk.2 B V
Assertion
Ref Expression
elssetk (⟪{A}, B SkA B)

Proof of Theorem elssetk
StepHypRef Expression
1 elssetk.1 . 2 A V
2 elssetk.2 . 2 B V
3 elssetkg 4269 . 2 ((A V B V) → (⟪{A}, B SkA B))
41, 2, 3mp2an 653 1 (⟪{A}, B SkA B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2859  {csn 3737  ⟪copk 4057   Sk cssetk 4183 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-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:  dfuni3  4315  dfint3  4318  ndisjrelk  4323  dfpw2  4327  dfaddc2  4381  dfnnc2  4395  nnsucelrlem1  4424  ltfinex  4464  ssfin  4470  eqpwrelk  4478  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  nnpweqlem1  4522  srelk  4524  tfinnnlem1  4533  spfinex  4537  dfop2lem1  4573  setconslem1  4731  setconslem2  4732  setconslem3  4733  setconslem7  4737  dfswap2  4741
 Copyright terms: Public domain W3C validator