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Theorem elxpi 4555
Description: Membership in a cross product. Uses fewer axioms than elxp 4556. (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 2146 . . . . . 6  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
21anbi1d 460 . . . . 5  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
322exbidv 1840 . . . 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 2830 . . 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 175 . 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 4545 . . 3  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
7 df-opab 3990 . . 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 2160 . 2  |-  ( B  X.  C )  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  B  /\  y  e.  C ) ) }
95, 8eleq2s 2234 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 103    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2125   <.cop 3530   {copab 3988    X. cxp 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-opab 3990  df-xp 4545
This theorem is referenced by:  xpsspw  4651  dmaddpqlem  7185  nqpi  7186  enq0ref  7241  nqnq0  7249  nq0nn  7250  cnm  7640  axaddcl  7672  axmulcl  7674
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