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Theorem elxp3 4890
Description: Membership in a Cartesian 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 4856 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 eqcom 2478 . . . 4  |-  ( <.
x ,  y >.  =  A  <->  A  =  <. x ,  y >. )
3 opelxp 4869 . . . 4  |-  ( <.
x ,  y >.  e.  ( B  X.  C
)  <->  ( x  e.  B  /\  y  e.  C ) )
42, 3anbi12i 711 . . 3  |-  ( (
<. x ,  y >.  =  A  /\  <. x ,  y >.  e.  ( B  X.  C ) )  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
542exbii 1727 . 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 260 1  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( <. x ,  y >.  =  A  /\  <. x ,  y
>.  e.  ( B  X.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   <.cop 3965    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845
This theorem is referenced by:  optocl  4916  unixp0  5377
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