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Theorem npss 3550
Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3455. (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 pm4.61 426 . . 3  |-  ( -.  ( A  C_  B  ->  A  =  B )  <-> 
( A  C_  B  /\  -.  A  =  B ) )
2 dfpss2 3525 . . 3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
31, 2bitr4i 252 . 2  |-  ( -.  ( A  C_  B  ->  A  =  B )  <-> 
A  C.  B )
43con1bii 331 1  |-  ( -.  A  C.  B  <->  ( A  C_  B  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    C_ wss 3412    C. wpss 3413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-ne 2643  df-pss 3428
This theorem is referenced by:  ttukeylem7  8771  canthp1lem2  8907  pgpfac1lem1  16666  lspsncv0  17319  obslbs  18250
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