Theorem opelxpd 4484

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 4483	. 2 |- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) )
4	1, 2, 3	syl2anc 404	1 |- ( ph -> <. A , B >. e. ( C X. D ) )

Syntax hints:   -> wi 4   e. wcel 1439   <.cop 3453   X. cxp 4450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-xp 4458

This theorem is referenced by:  strslfv2d  11597