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Theorem uneqdifeqim 3452
Description: Two ways that  A and  B can "partition"  C (when  A and  B don't overlap and  A is a part of  C). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
uneqdifeqim  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  C  -> 
( C  \  A
)  =  B ) )

Proof of Theorem uneqdifeqim
StepHypRef Expression
1 uncom 3224 . . . 4  |-  ( B  u.  A )  =  ( A  u.  B
)
2 eqtr 2158 . . . . . 6  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( B  u.  A )  =  C )
32eqcomd 2146 . . . . 5  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  C  =  ( B  u.  A ) )
4 difeq1 3191 . . . . . 6  |-  ( C  =  ( B  u.  A )  ->  ( C  \  A )  =  ( ( B  u.  A )  \  A
) )
5 difun2 3446 . . . . . 6  |-  ( ( B  u.  A ) 
\  A )  =  ( B  \  A
)
6 eqtr 2158 . . . . . . 7  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( C  \  A )  =  ( B  \  A ) )
7 incom 3272 . . . . . . . . . 10  |-  ( A  i^i  B )  =  ( B  i^i  A
)
87eqeq1i 2148 . . . . . . . . 9  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
9 disj3 3419 . . . . . . . . 9  |-  ( ( B  i^i  A )  =  (/)  <->  B  =  ( B  \  A ) )
108, 9bitri 183 . . . . . . . 8  |-  ( ( A  i^i  B )  =  (/)  <->  B  =  ( B  \  A ) )
11 eqtr 2158 . . . . . . . . . 10  |-  ( ( ( C  \  A
)  =  ( B 
\  A )  /\  ( B  \  A )  =  B )  -> 
( C  \  A
)  =  B )
1211expcom 115 . . . . . . . . 9  |-  ( ( B  \  A )  =  B  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1312eqcoms 2143 . . . . . . . 8  |-  ( B  =  ( B  \  A )  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1410, 13sylbi 120 . . . . . . 7  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  \  A )  =  ( B  \  A )  ->  ( C  \  A )  =  B ) )
156, 14syl5com 29 . . . . . 6  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( C  \  A )  =  B ) )
164, 5, 15sylancl 410 . . . . 5  |-  ( C  =  ( B  u.  A )  ->  (
( A  i^i  B
)  =  (/)  ->  ( C  \  A )  =  B ) )
173, 16syl 14 . . . 4  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( ( A  i^i  B )  =  (/)  ->  ( C  \  A )  =  B ) )
181, 17mpan 421 . . 3  |-  ( ( A  u.  B )  =  C  ->  (
( A  i^i  B
)  =  (/)  ->  ( C  \  A )  =  B ) )
1918com12 30 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  u.  B )  =  C  ->  ( C  \  A )  =  B ) )
2019adantl 275 1  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  C  -> 
( C  \  A
)  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    \ cdif 3072    u. cun 3073    i^i cin 3074    C_ wss 3075   (/)c0 3367
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368
This theorem is referenced by: (None)
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