Theorem opkelcnvk 4250

Description: Kuratowski ordered pair membership in a Kuratowski converse. (Contributed by SF, 14-Jan-2015.)

Hypotheses

Ref	Expression
opkelcnvk.1	⊢ A ∈ V
opkelcnvk.2	⊢ B ∈ V

Assertion

Ref	Expression
opkelcnvk	⊢ (⟪A, B⟫ ∈ ◡kC ↔ ⟪B, A⟫ ∈ C)

Proof of Theorem opkelcnvk

Step	Hyp	Ref	Expression
1		opkelcnvk.1	. 2 ⊢ A ∈ V
2		opkelcnvk.2	. 2 ⊢ B ∈ V
3		opkelcnvkg 4249	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (⟪A, B⟫ ∈ ◡kC ↔ ⟪B, A⟫ ∈ C))
4	1, 2, 3	mp2an 653	1 ⊢ (⟪A, B⟫ ∈ ◡kC ↔ ⟪B, A⟫ ∈ C)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2859  ⟪copk 4057  ^◡_kccnvk 4175

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-cnvk 4186

This theorem is referenced by:  opkelimagekg  4271  cnvkxpk  4276  cnvkexg  4286  dfidk2  4313  dfuni3  4315  dfint3  4318  nncaddccl  4419  nnsucelrlem1  4424  preaddccan2lem1  4454  ltfintrilem1  4465  ncfinlowerlem1  4482  eqtfinrelk  4486  oddfinex  4504  evenodddisjlem1  4515  nnpweqlem1  4522  sfintfinlem1  4531  tfinnnlem1  4533  vfinspclt  4552  dfop2lem1  4573  dfproj12  4576  dfproj22  4577  phialllem1  4616  setconslem1  4731  setconslem2  4732  setconslem4  4734