Theorem opelxpd 4784

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 4783	. 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 2205   <.cop 3694   X. cxp 4749

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-opab 4174  df-xp 4757

This theorem is referenced by:  opabssxpd  4788  elovimad  6096  suplocsrlemb  8123  seqvalcd  10827  ctiunctlemfo  13207  strslfv2d  13272  imasaddfnlemg  13544  imasaddflemg  13546  txcnp  15153  upxp  15154  txcnmpt  15155  uptx  15156  txdis1cn  15160  txlm  15161  lmcn2  15162  txhmeo  15201  comet  15381  txmetcnp  15400  dvaddxxbr  15583  dvmulxxbr  15584  dvcoapbr  15589  mpodvdsmulf1o  15875  wlkelvv  16361