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Theorem uneqdifeq 3718
Description: Two ways to say that  A and  B partition  C (when 
A and  B don't overlap and  A is a part of  C). (Contributed by FL, 17-Nov-2008.)
Assertion
Ref Expression
uneqdifeq  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  C  <->  ( C  \  A )  =  B ) )

Proof of Theorem uneqdifeq
StepHypRef Expression
1 uncom 3493 . . . . 5  |-  ( B  u.  A )  =  ( A  u.  B
)
2 eqtr 2455 . . . . . . 7  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( B  u.  A )  =  C )
32eqcomd 2443 . . . . . 6  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  C  =  ( B  u.  A ) )
4 difeq1 3460 . . . . . . 7  |-  ( C  =  ( B  u.  A )  ->  ( C  \  A )  =  ( ( B  u.  A )  \  A
) )
5 difun2 3709 . . . . . . 7  |-  ( ( B  u.  A ) 
\  A )  =  ( B  \  A
)
6 eqtr 2455 . . . . . . . 8  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( C  \  A )  =  ( B  \  A ) )
7 incom 3535 . . . . . . . . . . 11  |-  ( A  i^i  B )  =  ( B  i^i  A
)
87eqeq1i 2445 . . . . . . . . . 10  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
9 disj3 3674 . . . . . . . . . 10  |-  ( ( B  i^i  A )  =  (/)  <->  B  =  ( B  \  A ) )
108, 9bitri 242 . . . . . . . . 9  |-  ( ( A  i^i  B )  =  (/)  <->  B  =  ( B  \  A ) )
11 eqtr 2455 . . . . . . . . . . 11  |-  ( ( ( C  \  A
)  =  ( B 
\  A )  /\  ( B  \  A )  =  B )  -> 
( C  \  A
)  =  B )
1211expcom 426 . . . . . . . . . 10  |-  ( ( B  \  A )  =  B  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1312eqcoms 2441 . . . . . . . . 9  |-  ( B  =  ( B  \  A )  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1410, 13sylbi 189 . . . . . . . 8  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  \  A )  =  ( B  \  A )  ->  ( C  \  A )  =  B ) )
156, 14syl5com 29 . . . . . . 7  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( C  \  A )  =  B ) )
164, 5, 15sylancl 645 . . . . . 6  |-  ( C  =  ( B  u.  A )  ->  (
( A  i^i  B
)  =  (/)  ->  ( C  \  A )  =  B ) )
173, 16syl 16 . . . . 5  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( ( A  i^i  B )  =  (/)  ->  ( C  \  A )  =  B ) )
181, 17mpan 653 . . . 4  |-  ( ( A  u.  B )  =  C  ->  (
( A  i^i  B
)  =  (/)  ->  ( C  \  A )  =  B ) )
1918com12 30 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  u.  B )  =  C  ->  ( C  \  A )  =  B ) )
2019adantl 454 . 2  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  C  -> 
( C  \  A
)  =  B ) )
21 difss 3476 . . . . . . . 8  |-  ( C 
\  A )  C_  C
22 sseq1 3371 . . . . . . . . 9  |-  ( ( C  \  A )  =  B  ->  (
( C  \  A
)  C_  C  <->  B  C_  C
) )
23 unss 3523 . . . . . . . . . . 11  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
2423biimpi 188 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A  u.  B
)  C_  C )
2524expcom 426 . . . . . . . . 9  |-  ( B 
C_  C  ->  ( A  C_  C  ->  ( A  u.  B )  C_  C ) )
2622, 25syl6bi 221 . . . . . . . 8  |-  ( ( C  \  A )  =  B  ->  (
( C  \  A
)  C_  C  ->  ( A  C_  C  ->  ( A  u.  B ) 
C_  C ) ) )
2721, 26mpi 17 . . . . . . 7  |-  ( ( C  \  A )  =  B  ->  ( A  C_  C  ->  ( A  u.  B )  C_  C ) )
2827com12 30 . . . . . 6  |-  ( A 
C_  C  ->  (
( C  \  A
)  =  B  -> 
( A  u.  B
)  C_  C )
)
2928adantr 453 . . . . 5  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( C  \  A
)  =  B  -> 
( A  u.  B
)  C_  C )
)
3029imp 420 . . . 4  |-  ( ( ( A  C_  C  /\  ( A  i^i  B
)  =  (/) )  /\  ( C  \  A )  =  B )  -> 
( A  u.  B
)  C_  C )
31 eqimss 3402 . . . . . . 7  |-  ( ( C  \  A )  =  B  ->  ( C  \  A )  C_  B )
3231adantl 454 . . . . . 6  |-  ( ( A  C_  C  /\  ( C  \  A )  =  B )  -> 
( C  \  A
)  C_  B )
33 ssundif 3713 . . . . . 6  |-  ( C 
C_  ( A  u.  B )  <->  ( C  \  A )  C_  B
)
3432, 33sylibr 205 . . . . 5  |-  ( ( A  C_  C  /\  ( C  \  A )  =  B )  ->  C  C_  ( A  u.  B ) )
3534adantlr 697 . . . 4  |-  ( ( ( A  C_  C  /\  ( A  i^i  B
)  =  (/) )  /\  ( C  \  A )  =  B )  ->  C  C_  ( A  u.  B ) )
3630, 35eqssd 3367 . . 3  |-  ( ( ( A  C_  C  /\  ( A  i^i  B
)  =  (/) )  /\  ( C  \  A )  =  B )  -> 
( A  u.  B
)  =  C )
3736ex 425 . 2  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( C  \  A
)  =  B  -> 
( A  u.  B
)  =  C ) )
3820, 37impbid 185 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    <-> wb 178    /\ wa 360    = wceq 1653    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630
This theorem is referenced by:  hashbclem  11706  lecldbas  17288  conndisj  17484  ptuncnv  17844  ptunhmeo  17845  cldsubg  18145  icopnfcld  18807  iocmnfcld  18808  voliunlem1  19449  icombl  19463  ioombl  19464  uniioombllem4  19483  ismbf3d  19549  lhop  19905  subfacp1lem3  24873  subfacp1lem5  24875  pconcon  24923  cvmscld  24965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631
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