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Theorem elxpi 4886
Description: Membership in a cross product. Uses fewer axioms than elxp 4887. (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 2441 . . . . . 6  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
21anbi1d 686 . . . . 5  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
322exbidv 1638 . . . 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 3075 . . 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 233 . 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 4876 . . 3  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
7 df-opab 4259 . . 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 2455 . 2  |-  ( B  X.  C )  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  B  /\  y  e.  C ) ) }
95, 8eleq2s 2527 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 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   <.cop 3809   {copab 4257    X. cxp 4868
This theorem is referenced by:  xpsspw  4978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-opab 4259  df-xp 4876
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