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Theorem xp11 5448
Description: The Cartesian product of nonempty classes is one-to-one. (Contributed by NM, 31-May-2008.)
Assertion
Ref Expression
xp11  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  (
( A  X.  B
)  =  ( C  X.  D )  <->  ( A  =  C  /\  B  =  D ) ) )

Proof of Theorem xp11
StepHypRef Expression
1 xpnz 5432 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 anidm 644 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
3 neeq1 2748 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( A  X.  B
)  =/=  (/)  <->  ( C  X.  D )  =/=  (/) ) )
43anbi2d 703 . . . . . 6  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( ( A  X.  B )  =/=  (/)  /\  ( A  X.  B )  =/=  (/) )  <->  ( ( A  X.  B )  =/=  (/)  /\  ( C  X.  D )  =/=  (/) ) ) )
52, 4syl5bbr 259 . . . . 5  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( A  X.  B
)  =/=  (/)  <->  ( ( A  X.  B )  =/=  (/)  /\  ( C  X.  D )  =/=  (/) ) ) )
6 eqimss 3561 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( A  X.  B )  C_  ( C  X.  D
) )
7 ssxpb 5447 . . . . . . . 8  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  C_  ( C  X.  D
)  <->  ( A  C_  C  /\  B  C_  D
) ) )
86, 7syl5ibcom 220 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( A  X.  B
)  =/=  (/)  ->  ( A  C_  C  /\  B  C_  D ) ) )
9 eqimss2 3562 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( C  X.  D )  C_  ( A  X.  B
) )
10 ssxpb 5447 . . . . . . . 8  |-  ( ( C  X.  D )  =/=  (/)  ->  ( ( C  X.  D )  C_  ( A  X.  B
)  <->  ( C  C_  A  /\  D  C_  B
) ) )
119, 10syl5ibcom 220 . . . . . . 7  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( C  X.  D
)  =/=  (/)  ->  ( C  C_  A  /\  D  C_  B ) ) )
128, 11anim12d 563 . . . . . 6  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( ( A  X.  B )  =/=  (/)  /\  ( C  X.  D )  =/=  (/) )  ->  ( ( A  C_  C  /\  B  C_  D )  /\  ( C  C_  A  /\  D  C_  B ) ) ) )
13 an4 822 . . . . . . 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 ) ) )
14 eqss 3524 . . . . . . . 8  |-  ( A  =  C  <->  ( A  C_  C  /\  C  C_  A ) )
15 eqss 3524 . . . . . . . 8  |-  ( B  =  D  <->  ( B  C_  D  /\  D  C_  B ) )
1614, 15anbi12i 697 . . . . . . 7  |-  ( ( A  =  C  /\  B  =  D )  <->  ( ( A  C_  C  /\  C  C_  A )  /\  ( B  C_  D  /\  D  C_  B
) ) )
1713, 16bitr4i 252 . . . . . 6  |-  ( ( ( A  C_  C  /\  B  C_  D )  /\  ( C  C_  A  /\  D  C_  B
) )  <->  ( A  =  C  /\  B  =  D ) )
1812, 17syl6ib 226 . . . . 5  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( ( A  X.  B )  =/=  (/)  /\  ( C  X.  D )  =/=  (/) )  ->  ( A  =  C  /\  B  =  D ) ) )
195, 18sylbid 215 . . . 4  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  (
( A  X.  B
)  =/=  (/)  ->  ( A  =  C  /\  B  =  D )
) )
2019com12 31 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  ( ( A  X.  B )  =  ( C  X.  D
)  ->  ( A  =  C  /\  B  =  D ) ) )
211, 20sylbi 195 . 2  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  (
( A  X.  B
)  =  ( C  X.  D )  -> 
( A  =  C  /\  B  =  D ) ) )
22 xpeq12 5024 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  X.  B
)  =  ( C  X.  D ) )
2321, 22impbid1 203 1  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  ->  (
( A  X.  B
)  =  ( C  X.  D )  <->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    =/= wne 2662    C_ wss 3481   (/)c0 3790    X. cxp 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-dm 5015  df-rn 5016
This theorem is referenced by:  xpcan  5449  xpcan2  5450  fseqdom  8419  axcc2lem  8828
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