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Theorem uneqdifeq 3618
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 3395 . . . . 5  |-  ( B  u.  A )  =  ( A  u.  B
)
2 eqtr 2375 . . . . . . 7  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( B  u.  A )  =  C )
32eqcomd 2363 . . . . . 6  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  C  =  ( B  u.  A ) )
4 difeq1 3363 . . . . . . 7  |-  ( C  =  ( B  u.  A )  ->  ( C  \  A )  =  ( ( B  u.  A )  \  A
) )
5 difun2 3609 . . . . . . 7  |-  ( ( B  u.  A ) 
\  A )  =  ( B  \  A
)
6 eqtr 2375 . . . . . . . 8  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( C  \  A )  =  ( B  \  A ) )
7 incom 3437 . . . . . . . . . . 11  |-  ( A  i^i  B )  =  ( B  i^i  A
)
87eqeq1i 2365 . . . . . . . . . 10  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
9 disj3 3575 . . . . . . . . . 10  |-  ( ( B  i^i  A )  =  (/)  <->  B  =  ( B  \  A ) )
108, 9bitri 240 . . . . . . . . 9  |-  ( ( A  i^i  B )  =  (/)  <->  B  =  ( B  \  A ) )
11 eqtr 2375 . . . . . . . . . . 11  |-  ( ( ( C  \  A
)  =  ( B 
\  A )  /\  ( B  \  A )  =  B )  -> 
( C  \  A
)  =  B )
1211expcom 424 . . . . . . . . . 10  |-  ( ( B  \  A )  =  B  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1312eqcoms 2361 . . . . . . . . 9  |-  ( B  =  ( B  \  A )  ->  (
( C  \  A
)  =  ( B 
\  A )  -> 
( C  \  A
)  =  B ) )
1410, 13sylbi 187 . . . . . . . 8  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  \  A )  =  ( B  \  A )  ->  ( C  \  A )  =  B ) )
156, 14syl5com 26 . . . . . . 7  |-  ( ( ( C  \  A
)  =  ( ( B  u.  A ) 
\  A )  /\  ( ( B  u.  A )  \  A
)  =  ( B 
\  A ) )  ->  ( ( A  i^i  B )  =  (/)  ->  ( C  \  A )  =  B ) )
164, 5, 15sylancl 643 . . . . . 6  |-  ( C  =  ( B  u.  A )  ->  (
( A  i^i  B
)  =  (/)  ->  ( C  \  A )  =  B ) )
173, 16syl 15 . . . . 5  |-  ( ( ( B  u.  A
)  =  ( A  u.  B )  /\  ( A  u.  B
)  =  C )  ->  ( ( A  i^i  B )  =  (/)  ->  ( C  \  A )  =  B ) )
181, 17mpan 651 . . . 4  |-  ( ( A  u.  B )  =  C  ->  (
( A  i^i  B
)  =  (/)  ->  ( C  \  A )  =  B ) )
1918com12 27 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  u.  B )  =  C  ->  ( C  \  A )  =  B ) )
2019adantl 452 . 2  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  C  -> 
( C  \  A
)  =  B ) )
21 difss 3379 . . . . . . . 8  |-  ( C 
\  A )  C_  C
22 sseq1 3275 . . . . . . . . 9  |-  ( ( C  \  A )  =  B  ->  (
( C  \  A
)  C_  C  <->  B  C_  C
) )
23 unss 3425 . . . . . . . . . . 11  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
2423biimpi 186 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A  u.  B
)  C_  C )
2524expcom 424 . . . . . . . . 9  |-  ( B 
C_  C  ->  ( A  C_  C  ->  ( A  u.  B )  C_  C ) )
2622, 25syl6bi 219 . . . . . . . 8  |-  ( ( C  \  A )  =  B  ->  (
( C  \  A
)  C_  C  ->  ( A  C_  C  ->  ( A  u.  B ) 
C_  C ) ) )
2721, 26mpi 16 . . . . . . 7  |-  ( ( C  \  A )  =  B  ->  ( A  C_  C  ->  ( A  u.  B )  C_  C ) )
2827com12 27 . . . . . 6  |-  ( A 
C_  C  ->  (
( C  \  A
)  =  B  -> 
( A  u.  B
)  C_  C )
)
2928adantr 451 . . . . 5  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( C  \  A
)  =  B  -> 
( A  u.  B
)  C_  C )
)
3029imp 418 . . . 4  |-  ( ( ( A  C_  C  /\  ( A  i^i  B
)  =  (/) )  /\  ( C  \  A )  =  B )  -> 
( A  u.  B
)  C_  C )
31 eqimss 3306 . . . . . . 7  |-  ( ( C  \  A )  =  B  ->  ( C  \  A )  C_  B )
3231adantl 452 . . . . . 6  |-  ( ( A  C_  C  /\  ( C  \  A )  =  B )  -> 
( C  \  A
)  C_  B )
33 ssundif 3613 . . . . . 6  |-  ( C 
C_  ( A  u.  B )  <->  ( C  \  A )  C_  B
)
3432, 33sylibr 203 . . . . 5  |-  ( ( A  C_  C  /\  ( C  \  A )  =  B )  ->  C  C_  ( A  u.  B ) )
3534adantlr 695 . . . 4  |-  ( ( ( A  C_  C  /\  ( A  i^i  B
)  =  (/) )  /\  ( C  \  A )  =  B )  ->  C  C_  ( A  u.  B ) )
3630, 35eqssd 3272 . . 3  |-  ( ( ( A  C_  C  /\  ( A  i^i  B
)  =  (/) )  /\  ( C  \  A )  =  B )  -> 
( A  u.  B
)  =  C )
3736ex 423 . 2  |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  (
( C  \  A
)  =  B  -> 
( A  u.  B
)  =  C ) )
3820, 37impbid 183 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 176    /\ wa 358    = wceq 1642    \ cdif 3225    u. cun 3226    i^i cin 3227    C_ wss 3228   (/)c0 3531
This theorem is referenced by:  hashbclem  11486  lecldbas  17055  conndisj  17248  ptuncnv  17604  ptunhmeo  17605  cldsubg  17895  icopnfcld  18379  iocmnfcld  18380  voliunlem1  19011  icombl  19025  ioombl  19026  uniioombllem4  19045  ismbf3d  19113  lhop  19467  subfacp1lem3  24117  subfacp1lem5  24119  pconcon  24166  cvmscld  24208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532
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