Theorem opelxp1 5150

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 5146	. 2 |- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
2	1	simplbi 476	1 |- ( <. A , B >. e. ( C X. D ) -> A e. C )

Syntax hints:   -> wi 4   e. wcel 1990   <.cop 4183   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by:  otelxp1  5152  dff3  6372  ressnop0  6420  swoord1  7773  swoord2  7774  canthp1lem2  9475  ciclcl  16462  txcmplem1  21444  txlm  21451  dvbsss  23666  nvvcop  27449  nvvop  27464  prsdm  29960  linedegen  32250  opelopab3  33511