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Theorem elxp2 4723
Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
Assertion
Ref Expression
elxp2  |-  ( A  e.  ( B  X.  C )  <->  E. x  e.  B  E. y  e.  C  A  =  <. x ,  y >.
)
Distinct variable groups:    x, y, A    x, B, y    x, C, y

Proof of Theorem elxp2
StepHypRef Expression
1 df-rex 2562 . . . 4  |-  ( E. y  e.  C  ( x  e.  B  /\  A  =  <. x ,  y >. )  <->  E. y
( y  e.  C  /\  ( x  e.  B  /\  A  =  <. x ,  y >. )
) )
2 r19.42v 2707 . . . 4  |-  ( E. y  e.  C  ( x  e.  B  /\  A  =  <. x ,  y >. )  <->  ( x  e.  B  /\  E. y  e.  C  A  =  <. x ,  y >.
) )
3 an13 774 . . . . 5  |-  ( ( y  e.  C  /\  ( x  e.  B  /\  A  =  <. x ,  y >. )
)  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
43exbii 1572 . . . 4  |-  ( E. y ( y  e.  C  /\  ( x  e.  B  /\  A  =  <. x ,  y
>. ) )  <->  E. y
( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) )
51, 2, 43bitr3i 266 . . 3  |-  ( ( x  e.  B  /\  E. y  e.  C  A  =  <. x ,  y
>. )  <->  E. y ( A  =  <. x ,  y
>.  /\  ( x  e.  B  /\  y  e.  C ) ) )
65exbii 1572 . 2  |-  ( E. x ( x  e.  B  /\  E. y  e.  C  A  =  <. x ,  y >.
)  <->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) )
7 df-rex 2562 . 2  |-  ( E. x  e.  B  E. y  e.  C  A  =  <. x ,  y
>. 
<->  E. x ( x  e.  B  /\  E. y  e.  C  A  =  <. x ,  y
>. ) )
8 elxp 4722 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
96, 7, 83bitr4ri 269 1  |-  ( A  e.  ( B  X.  C )  <->  E. x  e.  B  E. y  e.  C  A  =  <. x ,  y >.
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656    X. cxp 4703
This theorem is referenced by:  opelxp  4735  xpiundi  4759  xpiundir  4760  xpdom2  6973  tskxpss  8410  nqereu  8569  elreal  8769  xpnnenOLD  12504  efgmnvl  15039  frgpuptinv  15096  frgpup3lem  15102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711
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