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Theorem elxpi 4705
Description: Membership in a cross product. Uses fewer axioms than elxp 4706. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxpi  |-  ( 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 elxpi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2289 . . . . . 6  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
21anbi1d 685 . . . . 5  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
322exbidv 1614 . . . 4  |-  ( z  =  A  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) ) )
43elabg 2915 . . 3  |-  ( A  e.  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) }  ->  ( A  e.  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) }  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
54ibi 232 . 2  |-  ( A  e.  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) }  ->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
6 df-xp 4695 . . 3  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
7 df-opab 4078 . . 3  |-  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) }
86, 7eqtri 2303 . 2  |-  ( B  X.  C )  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  B  /\  y  e.  C ) ) }
95, 8eleq2s 2375 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:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   <.cop 3643   {copab 4076    X. cxp 4687
This theorem is referenced by:  xpsspw  4797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-opab 4078  df-xp 4695
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