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Theorem elxpi 4444
Description: Membership in a cross product. Uses fewer axioms than elxp 4445. (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 2094 . . . . . 6  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
21anbi1d 453 . . . . 5  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
322exbidv 1796 . . . 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 2759 . . 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 174 . 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 4434 . . 3  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
7 df-opab 3892 . . 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 2108 . 2  |-  ( B  X.  C )  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  B  /\  y  e.  C ) ) }
95, 8eleq2s 2182 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 102    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   <.cop 3444   {copab 3890    X. cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-opab 3892  df-xp 4434
This theorem is referenced by:  xpsspw  4538  dmaddpqlem  6915  nqpi  6916  enq0ref  6971  nqnq0  6979  nq0nn  6980  axaddcl  7380  axmulcl  7382
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