Theorem opkelcnvk 4251

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 4250	. 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 2860  ⟪copk 4058  ^◡_kccnvk 4176

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-cnvk 4187

This theorem is referenced by:  opkelimagekg  4272  cnvkxpk  4277  cnvkexg  4287  dfidk2  4314  dfuni3  4316  dfint3  4319  nncaddccl  4420  nnsucelrlem1  4425  preaddccan2lem1  4455  ltfintrilem1  4466  ncfinlowerlem1  4483  eqtfinrelk  4487  oddfinex  4505  evenodddisjlem1  4516  nnpweqlem1  4523  sfintfinlem1  4532  tfinnnlem1  4534  vfinspclt  4553  dfop2lem1  4574  dfproj12  4577  dfproj22  4578  phialllem1  4617  setconslem1  4732  setconslem2  4733  setconslem4  4735