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Theorem xpmlem 5006
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 1912 . . 3  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  <->  ( E. x  x  e.  A  /\  E. y 
y  e.  B ) )
2 vex 2715 . . . . . 6  |-  x  e. 
_V
3 vex 2715 . . . . . 6  |-  y  e. 
_V
42, 3opex 4189 . . . . 5  |-  <. x ,  y >.  e.  _V
5 eleq1 2220 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
6 opelxp 4616 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
75, 6bitrdi 195 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
84, 7spcev 2807 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
98exlimivv 1876 . . 3  |-  ( E. x E. y ( x  e.  A  /\  y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
101, 9sylbir 134 . 2  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  ->  E. z  z  e.  ( A  X.  B
) )
11 elxp 4603 . . . . 5  |-  ( z  e.  ( A  X.  B )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
12 simpr 109 . . . . . 6  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( x  e.  A  /\  y  e.  B ) )
13122eximi 1581 . . . . 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 120 . . . 4  |-  ( z  e.  ( A  X.  B )  ->  E. x E. y ( x  e.  A  /\  y  e.  B ) )
1514exlimiv 1578 . . 3  |-  ( E. z  z  e.  ( A  X.  B )  ->  E. x E. y
( x  e.  A  /\  y  e.  B
) )
1615, 1sylib 121 . 2  |-  ( E. z  z  e.  ( A  X.  B )  ->  ( E. x  x  e.  A  /\  E. y  y  e.  B
) )
1710, 16impbii 125 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 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   <.cop 3563    X. cxp 4584
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-xp 4592
This theorem is referenced by:  xpm  5007
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