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Theorem npss 3286
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3194. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
npss  |-  ( -.  A  C.  B  <->  ( A  C_  B  ->  A  =  B ) )

Proof of Theorem npss
StepHypRef Expression
1 pm4.61 415 . . 3  |-  ( -.  ( A  C_  B  ->  A  =  B )  <-> 
( A  C_  B  /\  -.  A  =  B ) )
2 dfpss2 3261 . . 3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
31, 2bitr4i 243 . 2  |-  ( -.  ( A  C_  B  ->  A  =  B )  <-> 
A  C.  B )
43con1bii 321 1  |-  ( -.  A  C.  B  <->  ( A  C_  B  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  ttukeylem7  8142  canthp1lem2  8275  pgpfac1lem1  15309  lspsncv0  15899  obslbs  16630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-ne 2448  df-pss 3168
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