MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelxp1 Structured version   Unicode version

Theorem opelxp1 4891
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
StepHypRef Expression
1 opelxp 4888 . 2  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
21simplbi 460 1  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  A  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   <.cop 3902    X. cxp 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-opab 4370  df-xp 4865
This theorem is referenced by:  otelxp1  4893  dff3  5875  ressnop0  5908  swoord1  7149  swoord2  7150  canthp1lem2  8839  txcmplem1  19233  txlm  19240  dvbsss  21396  vcoprnelem  23975  nvvcop  23991  nvvop  24006  prsdm  26363  linedegen  28193  opelopab3  28633
  Copyright terms: Public domain W3C validator