Theorem opkelxpk 4248

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 4247	. 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 2859  ⟪copk 4057   ×_k cxpk 4174

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-xpk 4185

This theorem is referenced by:  elp6  4263  opkelimagekg  4271  cnvkxpk  4276  inxpk  4277  ins2kss  4279  ins3kss  4280  cokrelk  4284  cnvkexg  4286  dfuni12  4291  ssetkex  4294  sikexg  4296  dfimak2  4298  dfpw12  4301  ins2kexg  4305  ins3kexg  4306  dfpw2  4327  dfnnc2  4395  nnsucelrlem1  4424  ltfinex  4464  ssfin  4470  eqpw1relk  4479  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  srelk  4524  tfinnnlem1  4533  dfphi2  4569  dfop2lem1  4573  setconslem2  4732  setconslem4  4734  setconslem6  4736