MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pssdifcom2 Unicode version

Theorem pssdifcom2 3540
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 3309 . . . 4  |-  ( ( B  C_  C  /\  A  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
21ancoms 439 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( B  C_  ( C  \  A )  <->  A  C_  ( C  \  B ) ) )
3 difcom 3538 . . . . 5  |-  ( ( C  \  A ) 
C_  B  <->  ( C  \  B )  C_  A
)
43a1i 10 . . . 4  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( C  \  A )  C_  B  <->  ( C  \  B ) 
C_  A ) )
54notbid 285 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( -.  ( C 
\  A )  C_  B 
<->  -.  ( C  \  B )  C_  A
) )
62, 5anbi12d 691 . 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 3262 . 2  |-  ( B 
C.  ( C  \  A )  <->  ( B  C_  ( C  \  A
)  /\  -.  ( C  \  A )  C_  B ) )
8 dfpss3 3262 . 2  |-  ( A 
C.  ( C  \  B )  <->  ( A  C_  ( C  \  B
)  /\  -.  ( C  \  B )  C_  A ) )
96, 7, 83bitr4g 279 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 176    /\ wa 358    \ cdif 3149    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  fin2i2  7944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168
  Copyright terms: Public domain W3C validator