ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneqdifeqim Unicode version

Theorem uneqdifeqim 3335
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 3117 . . . 4  |-  ( B  u.  A )  =  ( A  u.  B
)
2 eqtr 2099 . . . . . 6  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( B  u.  A )  =  C )
32eqcomd 2087 . . . . 5  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  C  =  ( B  u.  A ) )
4 difeq1 3084 . . . . . 6  |-  ( C  =  ( B  u.  A )  ->  ( C  \  A )  =  ( ( B  u.  A )  \  A
) )
5 difun2 3329 . . . . . 6  |-  ( ( B  u.  A ) 
\  A )  =  ( B  \  A
)
6 eqtr 2099 . . . . . . 7  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( C  \  A )  =  ( B  \  A ) )
7 incom 3165 . . . . . . . . . 10  |-  ( A  i^i  B )  =  ( B  i^i  A
)
87eqeq1i 2089 . . . . . . . . 9  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
9 disj3 3303 . . . . . . . . 9  |-  ( ( B  i^i  A )  =  (/)  <->  B  =  ( B  \  A ) )
108, 9bitri 182 . . . . . . . 8  |-  ( ( A  i^i  B )  =  (/)  <->  B  =  ( B  \  A ) )
11 eqtr 2099 . . . . . . . . . 10  |-  ( ( ( C  \  A
)  =  ( B 
\  A )  /\  ( B  \  A )  =  B )  -> 
( C  \  A
)  =  B )
1211expcom 114 . . . . . . . . 9  |-  ( ( B  \  A )  =  B  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1312eqcoms 2085 . . . . . . . 8  |-  ( B  =  ( B  \  A )  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1410, 13sylbi 119 . . . . . . 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 404 . . . . 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 415 . . 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 271 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 102    = wceq 1285    \ cdif 2971    u. cun 2972    i^i cin 2973    C_ wss 2974   (/)c0 3258
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-in1 577  ax-in2 578  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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator