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Theorem dfpss3 3275
Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfpss3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )

Proof of Theorem dfpss3
StepHypRef Expression
1 dfpss2 3274 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
2 eqss 3207 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
32baib 871 . . . 4  |-  ( A 
C_  B  ->  ( A  =  B  <->  B  C_  A
) )
43notbid 285 . . 3  |-  ( A 
C_  B  ->  ( -.  A  =  B  <->  -.  B  C_  A )
)
54pm5.32i 618 . 2  |-  ( ( A  C_  B  /\  -.  A  =  B
)  <->  ( A  C_  B  /\  -.  B  C_  A ) )
61, 5bitri 240 1  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1632    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  pssirr  3289  pssn2lp  3290  ssnpss  3292  nsspssun  3415  npss0  3506  pssdifcom1  3552  pssdifcom2  3553  php3  7063  fincssdom  7965  reclem2pr  8688  islbs3  15924  chpsscon3  22098  chpssati  22959  fundmpss  24193  vxveqv  25157  lpssat  29825  lssat  29828  dihglblem6  32152
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179  df-pss 3181
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