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Theorem dfpss3 3262
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 3261 . 2  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
2 eqss 3194 . . . . 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 1623    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  pssirr  3276  pssn2lp  3277  ssnpss  3279  nsspssun  3402  npss0  3493  pssdifcom1  3539  pssdifcom2  3540  php3  7047  fincssdom  7949  reclem2pr  8672  islbs3  15908  chpsscon3  22082  chpssati  22943  fundmpss  24122  vxveqv  25054  lpssat  29203  lssat  29206  dihglblem6  31530
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-in 3159  df-ss 3166  df-pss 3168
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