Theorem opelxpd 4758

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 4757	. 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 2202   <.cop 3672   X. cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731

This theorem is referenced by:  opabssxpd  4762  elovimad  6061  suplocsrlemb  8025  seqvalcd  10722  ctiunctlemfo  13059  strslfv2d  13124  imasaddfnlemg  13396  imasaddflemg  13398  txcnp  14994  upxp  14995  txcnmpt  14996  uptx  14997  txdis1cn  15001  txlm  15002  lmcn2  15003  txhmeo  15042  comet  15222  txmetcnp  15241  dvaddxxbr  15424  dvmulxxbr  15425  dvcoapbr  15430  mpodvdsmulf1o  15713  wlkelvv  16199