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Theorem xpmlem 4794
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
Assertion
Ref Expression
xpmlem  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  <->  E. z  z  e.  ( A  X.  B
) )
Distinct variable groups:    x, y, z, A    x, B, y, z

Proof of Theorem xpmlem
StepHypRef Expression
1 eeanv 1850 . . 3  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  <->  ( E. x  x  e.  A  /\  E. y 
y  e.  B ) )
2 vex 2613 . . . . . 6  |-  x  e. 
_V
3 vex 2613 . . . . . 6  |-  y  e. 
_V
42, 3opex 4012 . . . . 5  |-  <. x ,  y >.  e.  _V
5 eleq1 2145 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
6 opelxp 4420 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
75, 6syl6bb 194 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
84, 7spcev 2701 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
98exlimivv 1819 . . 3  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
101, 9sylbir 133 . 2  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
11 elxp 4408 . . . . 5  |-  ( z  e.  ( A  X.  B )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
12 simpr 108 . . . . . 6  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( x  e.  A  /\  y  e.  B ) )
13122eximi 1533 . . . . 5  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  E. x E. y ( x  e.  A  /\  y  e.  B ) )
1411, 13sylbi 119 . . . 4  |-  ( z  e.  ( A  X.  B )  ->  E. x E. y ( x  e.  A  /\  y  e.  B ) )
1514exlimiv 1530 . . 3  |-  ( E. z  z  e.  ( A  X.  B )  ->  E. x E. y
( x  e.  A  /\  y  e.  B
) )
1615, 1sylib 120 . 2  |-  ( E. z  z  e.  ( A  X.  B )  ->  ( E. x  x  e.  A  /\  E. y  y  e.  B
) )
1710, 16impbii 124 1  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  <->  E. z  z  e.  ( A  X.  B
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   <.cop 3419    X. cxp 4389
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-xp 4397
This theorem is referenced by:  xpm  4795
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