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Theorem xpm 4821
Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
Assertion
Ref Expression
xpm  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  <->  E. z  z  e.  ( A  X.  B
) )
Distinct variable groups:    x, A    y, B    z, A    z, B
Allowed substitution hints:    A( y)    B( x)

Proof of Theorem xpm
Dummy variables  a  b  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpmlem 4820 . 2  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <->  E. w  w  e.  ( A  X.  B
) )
2 eleq1 2147 . . . 4  |-  ( a  =  x  ->  (
a  e.  A  <->  x  e.  A ) )
32cbvexv 1840 . . 3  |-  ( E. a  a  e.  A  <->  E. x  x  e.  A
)
4 eleq1 2147 . . . 4  |-  ( b  =  y  ->  (
b  e.  B  <->  y  e.  B ) )
54cbvexv 1840 . . 3  |-  ( E. b  b  e.  B  <->  E. y  y  e.  B
)
63, 5anbi12i 448 . 2  |-  ( ( E. a  a  e.  A  /\  E. b 
b  e.  B )  <-> 
( E. x  x  e.  A  /\  E. y  y  e.  B
) )
7 eleq1 2147 . . 3  |-  ( w  =  z  ->  (
w  e.  ( A  X.  B )  <->  z  e.  ( A  X.  B
) ) )
87cbvexv 1840 . 2  |-  ( E. w  w  e.  ( A  X.  B )  <->  E. z  z  e.  ( A  X.  B
) )
91, 6, 83bitr3i 208 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   E.wex 1424    e. wcel 1436    X. cxp 4411
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3877  df-xp 4419
This theorem is referenced by:  ssxpbm  4834  xp11m  4837  xpexr2m  4840  unixpm  4934
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