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

Theorem elxp 4898
Description: Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxp  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y

Proof of Theorem elxp
StepHypRef Expression
1 df-xp 4887 . . 3  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
21eleq2i 2502 . 2  |-  ( A  e.  ( B  X.  C )  <->  A  e.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )
3 elopab 4465 . 2  |-  ( A  e.  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) )
42, 3bitri 242 1  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   <.cop 3819   {copab 4268    X. cxp 4879
This theorem is referenced by:  elxp2  4899  0nelxp  4909  0nelelxp  4910  rabxp  4917  elxp3  4931  elvv  4939  elvvv  4940  xp0r  4959  elxp4  5360  elxp5  5361  dfco2a  5373  opabex3d  5992  opabex3  5993  xp1st  6379  xp2nd  6380  poxp  6461  soxp  6462  xpsnen  7195  xpcomco  7201  xpassen  7205  dfac5lem1  8009  dfac5lem4  8012  axdc4lem  8340  fsum2dlem  12559  fprod2dlem  25309  dfres3  25387  elima4  25409  brcart  25782  brimg  25787  dibelval3  32019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887
  Copyright terms: Public domain W3C validator