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Theorem pssdifcom2 3919
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 3642 . . . 4  |-  ( ( B  C_  C  /\  A  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
21ancoms 453 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
3 difcom 3917 . . . . 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 294 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( -.  ( C 
\  A )  C_  B 
<->  -.  ( C  \  B )  C_  A
) )
62, 5anbi12d 710 . 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 3595 . 2  |-  ( B 
C.  ( C  \  A )  <->  ( B  C_  ( C  \  A
)  /\  -.  ( C  \  A )  C_  B ) )
8 dfpss3 3595 . 2  |-  ( A 
C.  ( C  \  B )  <->  ( A  C_  ( C  \  B
)  /\  -.  ( C  \  B )  C_  A ) )
96, 7, 83bitr4g 288 1  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C.  ( C  \  A )  <->  A  C.  ( C  \  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \ cdif 3478    C_ wss 3481    C. wpss 3482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497
This theorem is referenced by:  fin2i2  8710
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