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Theorem pssdifcom2 3714
Description: Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
Assertion
Ref Expression
pssdifcom2  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C.  ( C  \  A )  <->  A  C.  ( C  \  B ) ) )

Proof of Theorem pssdifcom2
StepHypRef Expression
1 ssconb 3480 . . . 4  |-  ( ( B  C_  C  /\  A  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
21ancoms 440 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
3 difcom 3712 . . . . 5  |-  ( ( C  \  A ) 
C_  B  <->  ( C  \  B )  C_  A
)
43a1i 11 . . . 4  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( C  \  A )  C_  B  <->  ( C  \  B ) 
C_  A ) )
54notbid 286 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( -.  ( C 
\  A )  C_  B 
<->  -.  ( C  \  B )  C_  A
) )
62, 5anbi12d 692 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( B  C_  ( C  \  A )  /\  -.  ( C 
\  A )  C_  B )  <->  ( A  C_  ( C  \  B
)  /\  -.  ( C  \  B )  C_  A ) ) )
7 dfpss3 3433 . 2  |-  ( B 
C.  ( C  \  A )  <->  ( B  C_  ( C  \  A
)  /\  -.  ( C  \  A )  C_  B ) )
8 dfpss3 3433 . 2  |-  ( A 
C.  ( C  \  B )  <->  ( A  C_  ( C  \  B
)  /\  -.  ( C  \  B )  C_  A ) )
96, 7, 83bitr4g 280 1  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C.  ( C  \  A )  <->  A  C.  ( C  \  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    \ cdif 3317    C_ wss 3320    C. wpss 3321
This theorem is referenced by:  fin2i2  8198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336
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