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Theorem pssdifcom1 3552
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 3551 . . . 4  |-  ( ( C  \  A ) 
C_  B  <->  ( C  \  B )  C_  A
)
21a1i 10 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( C  \  A )  C_  B  <->  ( C  \  B ) 
C_  A ) )
3 ssconb 3322 . . . . 5  |-  ( ( B  C_  C  /\  A  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
43ancoms 439 . . . 4  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
54notbid 285 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( -.  B  C_  ( C  \  A )  <->  -.  A  C_  ( C 
\  B ) ) )
62, 5anbi12d 691 . 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 3275 . 2  |-  ( ( C  \  A ) 
C.  B  <->  ( ( C  \  A )  C_  B  /\  -.  B  C_  ( C  \  A ) ) )
8 dfpss3 3275 . 2  |-  ( ( C  \  B ) 
C.  A  <->  ( ( C  \  B )  C_  A  /\  -.  A  C_  ( C  \  B ) ) )
96, 7, 83bitr4g 279 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 176    /\ wa 358    \ cdif 3162    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  isfin2-2  7961  compssiso  8016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181
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