Theorem opelxp1 5038

Description: The first member of an ordered pair of classes in a Cartesian product belongs to first Cartesian 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 5035	. 2 |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
2	1	simplbi 460	1 |- ( <. A , B >. e. ( C X. D ) -> A e. C )

Syntax hints:   -> wi 4   e. wcel 1767   <.cop 4039   X. cxp 5003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-xp 5011

This theorem is referenced by:  otelxp1  5040  dff3  6045  ressnop0  6079  swoord1  7352  swoord2  7353  canthp1lem2  9043  txcmplem1  20010  txlm  20017  dvbsss  22174  vcoprnelem  25294  nvvcop  25310  nvvop  25325  prsdm  27721  linedegen  29720  opelopab3  30134