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Theorem xp11 5273
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 5257 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
2 anidm 644 . . . . . 6  |-  ( ( ( A  X.  B
)  =/=  (/)  /\  ( A  X.  B )  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
3 neeq1 2616 . . . . . . 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 3408 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( A  X.  B )  C_  ( C  X.  D
) )
7 ssxpb 5272 . . . . . . . 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 3409 . . . . . . . 8  |-  ( ( A  X.  B )  =  ( C  X.  D )  ->  ( C  X.  D )  C_  ( A  X.  B
) )
10 ssxpb 5272 . . . . . . . 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 820 . . . . . . 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 3371 . . . . . . . 8  |-  ( A  =  C  <->  ( A  C_  C  /\  C  C_  A ) )
15 eqss 3371 . . . . . . . 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 4859 . 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 1369    =/= wne 2606    C_ wss 3328   (/)c0 3637    X. cxp 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-cnv 4848  df-dm 4850  df-rn 4851
This theorem is referenced by:  xpcan  5274  xpcan2  5275  fseqdom  8196  axcc2lem  8605
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