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Theorem xpid11m 4585
Description: The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
Assertion
Ref Expression
xpid11m  |-  ( ( E. x  x  e.  A  /\  E. x  x  e.  B )  ->  ( ( A  X.  A )  =  ( B  X.  B )  <-> 
A  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem xpid11m
StepHypRef Expression
1 dmxpm 4583 . . . . . 6  |-  ( E. x  x  e.  A  ->  dom  ( A  X.  A )  =  A )
21adantr 270 . . . . 5  |-  ( ( E. x  x  e.  A  /\  E. x  x  e.  B )  ->  dom  ( A  X.  A )  =  A )
3 dmeq 4563 . . . . 5  |-  ( ( A  X.  A )  =  ( B  X.  B )  ->  dom  ( A  X.  A
)  =  dom  ( B  X.  B ) )
42, 3sylan9req 2135 . . . 4  |-  ( ( ( E. x  x  e.  A  /\  E. x  x  e.  B
)  /\  ( A  X.  A )  =  ( B  X.  B ) )  ->  A  =  dom  ( B  X.  B
) )
5 dmxpm 4583 . . . . 5  |-  ( E. x  x  e.  B  ->  dom  ( B  X.  B )  =  B )
65ad2antlr 473 . . . 4  |-  ( ( ( E. x  x  e.  A  /\  E. x  x  e.  B
)  /\  ( A  X.  A )  =  ( B  X.  B ) )  ->  dom  ( B  X.  B )  =  B )
74, 6eqtrd 2114 . . 3  |-  ( ( ( E. x  x  e.  A  /\  E. x  x  e.  B
)  /\  ( A  X.  A )  =  ( B  X.  B ) )  ->  A  =  B )
87ex 113 . 2  |-  ( ( E. x  x  e.  A  /\  E. x  x  e.  B )  ->  ( ( A  X.  A )  =  ( B  X.  B )  ->  A  =  B ) )
9 xpeq12 4390 . . 3  |-  ( ( A  =  B  /\  A  =  B )  ->  ( A  X.  A
)  =  ( B  X.  B ) )
109anidms 389 . 2  |-  ( A  =  B  ->  ( A  X.  A )  =  ( B  X.  B
) )
118, 10impbid1 140 1  |-  ( ( E. x  x  e.  A  /\  E. x  x  e.  B )  ->  ( ( A  X.  A )  =  ( B  X.  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    X. cxp 4369   dom cdm 4371
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-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-dm 4381
This theorem is referenced by: (None)
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