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Theorem pssdifcom1 3715
Description: Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
Assertion
Ref Expression
pssdifcom1  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( C  \  A )  C.  B  <->  ( C  \  B ) 
C.  A ) )

Proof of Theorem pssdifcom1
StepHypRef Expression
1 difcom 3714 . . . 4  |-  ( ( C  \  A ) 
C_  B  <->  ( C  \  B )  C_  A
)
21a1i 11 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( C  \  A )  C_  B  <->  ( C  \  B ) 
C_  A ) )
3 ssconb 3482 . . . . 5  |-  ( ( B  C_  C  /\  A  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
43ancoms 441 . . . 4  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
54notbid 287 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( -.  B  C_  ( C  \  A )  <->  -.  A  C_  ( C 
\  B ) ) )
62, 5anbi12d 693 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( ( C 
\  A )  C_  B  /\  -.  B  C_  ( C  \  A ) )  <->  ( ( C 
\  B )  C_  A  /\  -.  A  C_  ( C  \  B ) ) ) )
7 dfpss3 3435 . 2  |-  ( ( C  \  A ) 
C.  B  <->  ( ( C  \  A )  C_  B  /\  -.  B  C_  ( C  \  A ) ) )
8 dfpss3 3435 . 2  |-  ( ( C  \  B ) 
C.  A  <->  ( ( C  \  B )  C_  A  /\  -.  A  C_  ( C  \  B ) ) )
96, 7, 83bitr4g 281 1  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( C  \  A )  C.  B  <->  ( C  \  B ) 
C.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    \ cdif 3319    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  isfin2-2  8201  compssiso  8256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338
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