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Theorem xp11m 4789
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 4775 . . 3  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  <->  E. z  z  e.  ( A  X.  B
) )
2 anidm 388 . . . . . 6  |-  ( ( E. z  z  e.  ( A  X.  B
)  /\  E. z 
z  e.  ( A  X.  B ) )  <->  E. z  z  e.  ( A  X.  B
) )
3 eleq2 2143 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
z  e.  ( A  X.  B )  <->  z  e.  ( C  X.  D
) ) )
43exbidv 1747 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( A  X.  B
)  <->  E. z  z  e.  ( C  X.  D
) ) )
54anbi2d 452 . . . . . 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, 5syl5bbr 192 . . . . 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 3052 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( A  X.  B )  C_  ( C  X.  D
) )
8 ssxpbm 4786 . . . . . . . 8  |-  ( E. z  z  e.  ( A  X.  B )  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )
97, 8syl5ibcom 153 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( A  X.  B
)  ->  ( A  C_  C  /\  B  C_  D ) ) )
10 eqimss2 3053 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( C  X.  D )  C_  ( A  X.  B
) )
11 ssxpbm 4786 . . . . . . . 8  |-  ( E. z  z  e.  ( C  X.  D )  ->  ( ( C  X.  D )  C_  ( A  X.  B
)  <->  ( C  C_  A  /\  D  C_  B
) ) )
1210, 11syl5ibcom 153 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( E. z  z  e.  ( C  X.  D
)  ->  ( C  C_  A  /\  D  C_  B ) ) )
139, 12anim12d 328 . . . . . 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 551 . . . . . . 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 3015 . . . . . . . 8  |-  ( A  =  C  <->  ( A  C_  C  /\  C  C_  A ) )
16 eqss 3015 . . . . . . . 8  |-  ( B  =  D  <->  ( B  C_  D  /\  D  C_  B ) )
1715, 16anbi12i 448 . . . . . . 7  |-  ( ( A  =  C  /\  B  =  D )  <->  ( ( A  C_  C  /\  C  C_  A )  /\  ( B  C_  D  /\  D  C_  B
) ) )
1814, 17bitr4i 185 . . . . . 6  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  C_  A  /\  D  C_  B
) )  <->  ( A  =  C  /\  B  =  D ) )
1913, 18syl6ib 159 . . . . 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 148 . . . 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 119 . 2  |-  ( ( E. x  x  e.  A  /\  E. y 
y  e.  B )  ->  ( ( A  X.  B )  =  ( C  X.  D
)  ->  ( A  =  C  /\  B  =  D ) ) )
23 xpeq12 4390 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  X.  B
)  =  ( C  X.  D ) )
2422, 23impbid1 140 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 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2974    X. cxp 4369
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by: (None)
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