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Theorem xp11m 4977
Description: The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)
Assertion
Ref Expression
xp11m  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  ->  ( ( A  X.  B )  =  ( C  X.  D
)  <->  ( A  =  C  /\  B  =  D ) ) )
Distinct variable groups:    x, A    y, B
Allowed substitution hints:    A( y)    B( x)    C( x, y)    D( x, y)

Proof of Theorem xp11m
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 xpm 4960 . . 3  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  <->  E. z  z  e.  ( A  X.  B
) )
2 anidm 393 . . . . . 6  |-  ( ( E. z  z  e.  ( A  X.  B
)  /\  E. z 
z  e.  ( A  X.  B ) )  <->  E. z  z  e.  ( A  X.  B
) )
3 eleq2 2203 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
z  e.  ( A  X.  B )  <->  z  e.  ( C  X.  D
) ) )
43exbidv 1797 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( A  X.  B
)  <->  E. z  z  e.  ( C  X.  D
) ) )
54anbi2d 459 . . . . . 6  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( E. z  z  e.  ( A  X.  B )  /\  E. z  z  e.  ( A  X.  B ) )  <-> 
( E. z  z  e.  ( A  X.  B )  /\  E. z  z  e.  ( C  X.  D ) ) ) )
62, 5bitr3id 193 . . . . 5  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( A  X.  B
)  <->  ( E. z 
z  e.  ( A  X.  B )  /\  E. z  z  e.  ( C  X.  D ) ) ) )
7 eqimss 3151 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( A  X.  B )  C_  ( C  X.  D
) )
8 ssxpbm 4974 . . . . . . . 8  |-  ( E. z  z  e.  ( A  X.  B )  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )
97, 8syl5ibcom 154 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( A  X.  B
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
10 eqimss2 3152 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( C  X.  D )  C_  ( A  X.  B
) )
11 ssxpbm 4974 . . . . . . . 8  |-  ( E. z  z  e.  ( C  X.  D )  ->  ( ( C  X.  D )  C_  ( A  X.  B
)  <->  ( C  C_  A  /\  D  C_  B
) ) )
1210, 11syl5ibcom 154 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( C  X.  D
)  ->  ( C  C_  A  /\  D  C_  B ) ) )
139, 12anim12d 333 . . . . . 6  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( E. z  z  e.  ( A  X.  B )  /\  E. z  z  e.  ( C  X.  D ) )  ->  ( ( A 
C_  C  /\  B  C_  D )  /\  ( C  C_  A  /\  D  C_  B ) ) ) )
14 an4 575 . . . . . . 7  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  C_  A  /\  D  C_  B
) )  <->  ( ( A  C_  C  /\  C  C_  A )  /\  ( B  C_  D  /\  D  C_  B ) ) )
15 eqss 3112 . . . . . . . 8  |-  ( A  =  C  <->  ( A  C_  C  /\  C  C_  A ) )
16 eqss 3112 . . . . . . . 8  |-  ( B  =  D  <->  ( B  C_  D  /\  D  C_  B ) )
1715, 16anbi12i 455 . . . . . . 7  |-  ( ( A  =  C  /\  B  =  D )  <->  ( ( A  C_  C  /\  C  C_  A )  /\  ( B  C_  D  /\  D  C_  B
) ) )
1814, 17bitr4i 186 . . . . . 6  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  C_  A  /\  D  C_  B
) )  <->  ( A  =  C  /\  B  =  D ) )
1913, 18syl6ib 160 . . . . 5  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( E. z  z  e.  ( A  X.  B )  /\  E. z  z  e.  ( C  X.  D ) )  ->  ( A  =  C  /\  B  =  D ) ) )
206, 19sylbid 149 . . . 4  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( A  X.  B
)  ->  ( A  =  C  /\  B  =  D ) ) )
2120com12 30 . . 3  |-  ( E. z  z  e.  ( A  X.  B )  ->  ( ( A  X.  B )  =  ( C  X.  D
)  ->  ( A  =  C  /\  B  =  D ) ) )
221, 21sylbi 120 . 2  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  ->  ( ( A  X.  B )  =  ( C  X.  D
)  ->  ( A  =  C  /\  B  =  D ) ) )
23 xpeq12 4558 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  X.  B
)  =  ( C  X.  D ) )
2422, 23impbid1 141 1  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  ->  ( ( A  X.  B )  =  ( C  X.  D
)  <->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480    C_ wss 3071    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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  cc2lem  7081
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