Theorem opelxp1 4573

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 4569	. 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 1480   <.cop 3530   X. cxp 4537

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545

This theorem is referenced by:  otelxp1  4575  dmxpss  4969  nfvres  5454  ressnop0  5601  swoord1  6458  swoord2  6459  txlm  12448