Theorem opkelxpk 4249

Description: Kuratowski ordered pair membership in a Kuratowski cross product. (Contributed by SF, 13-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
opkelxpk	⊢ (⟪A, B⟫ ∈ (C ×k D) ↔ (A ∈ C ∧ B ∈ D))

Proof of Theorem opkelxpk

Step	Hyp	Ref	Expression
1		opkelxpk.1	. 2 ⊢ A ∈ V
2		opkelxpk.2	. 2 ⊢ B ∈ V
3		opkelxpkg 4248	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (⟪A, B⟫ ∈ (C ×k D) ↔ (A ∈ C ∧ B ∈ D)))
4	1, 2, 3	mp2an 653	1 ⊢ (⟪A, B⟫ ∈ (C ×k D) ↔ (A ∈ C ∧ B ∈ D))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Vcvv 2860  ⟪copk 4058   ×_k cxpk 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 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-xpk 4186

This theorem is referenced by:  elp6  4264  opkelimagekg  4272  cnvkxpk  4277  inxpk  4278  ins2kss  4280  ins3kss  4281  cokrelk  4285  cnvkexg  4287  dfuni12  4292  ssetkex  4295  sikexg  4297  dfimak2  4299  dfpw12  4302  ins2kexg  4306  ins3kexg  4307  dfpw2  4328  dfnnc2  4396  nnsucelrlem1  4425  ltfinex  4465  ssfin  4471  eqpw1relk  4480  ncfinraiselem2  4481  ncfinlowerlem1  4483  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  srelk  4525  tfinnnlem1  4534  dfphi2  4570  dfop2lem1  4574  setconslem2  4733  setconslem4  4735  setconslem6  4737