Theorem opelxpd 4696

Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
opelxpd.1	|- ( ph -> A e. C )
opelxpd.2	|- ( ph -> B e. D )

Assertion

Ref	Expression
opelxpd	|- ( ph -> <. A , B >. e. ( C X. D ) )

Proof of Theorem opelxpd

Step	Hyp	Ref	Expression
1		opelxpd.1	. 2 |- ( ph -> A e. C )
2		opelxpd.2	. 2 |- ( ph -> B e. D )
3		opelxpi 4695	. 2 |- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> <. A , B >. e. ( C X. D ) )

Syntax hints:   -> wi 4   e. wcel 2167   <.cop 3625   X. cxp 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669

This theorem is referenced by:  suplocsrlemb  7873  seqvalcd  10553  ctiunctlemfo  12656  strslfv2d  12721  imasaddfnlemg  12957  imasaddflemg  12959  txcnp  14507  upxp  14508  txcnmpt  14509  uptx  14510  txdis1cn  14514  txlm  14515  lmcn2  14516  txhmeo  14555  comet  14735  txmetcnp  14754  dvaddxxbr  14937  dvmulxxbr  14938  dvcoapbr  14943  mpodvdsmulf1o  15226