Theorem opelxp1 4621

Description: The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opelxp1	|- ( <. A , B >. e. ( C X. D ) -> A e. C )

Proof of Theorem opelxp1

Step	Hyp	Ref	Expression
1		opelxp 4617	. 2 |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
2	1	simplbi 272	1 |- ( <. A , B >. e. ( C X. D ) -> A e. C )

Syntax hints:   -> wi 4   e. wcel 2128   <.cop 3563   X. cxp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4027  df-xp 4593

This theorem is referenced by:  otelxp1  4623  dmxpss  5017  nfvres  5502  ressnop0  5649  swoord1  6510  swoord2  6511  txlm  12721