MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elxp3 Unicode version

Theorem elxp3 4868
Description: Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
Assertion
Ref Expression
elxp3  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( <. x ,  y >.  =  A  /\  <. x ,  y
>.  e.  ( B  X.  C ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y

Proof of Theorem elxp3
StepHypRef Expression
1 elxp 4835 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 eqcom 2389 . . . 4  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
3 opelxp 4848 . . . 4  |-  ( <.
x ,  y >.  e.  ( B  X.  C
)  <->  ( x  e.  B  /\  y  e.  C ) )
42, 3anbi12i 679 . . 3  |-  ( (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
542exbii 1590 . 2  |-  ( E. x E. y (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  <->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) )
61, 5bitr4i 244 1  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( <. x ,  y >.  =  A  /\  <. x ,  y
>.  e.  ( B  X.  C ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   <.cop 3760    X. cxp 4816
This theorem is referenced by:  optocl  4892  unixp0  5343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-opab 4208  df-xp 4824
  Copyright terms: Public domain W3C validator