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Theorem difsnpss 3758
Description:  ( B  \  { A } ) is a proper subclass of  B if and only if  A is a member of  B. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
difsnpss  |-  ( A  e.  B  <->  ( B  \  { A } ) 
C.  B )

Proof of Theorem difsnpss
StepHypRef Expression
1 notnot 282 . 2  |-  ( A  e.  B  <->  -.  -.  A  e.  B )
2 difss 3303 . . . 4  |-  ( B 
\  { A }
)  C_  B
32biantrur 492 . . 3  |-  ( ( B  \  { A } )  =/=  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
4 difsneq 3757 . . . 4  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
54necon3bbii 2477 . . 3  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  =/=  B
)
6 df-pss 3168 . . 3  |-  ( ( B  \  { A } )  C.  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
73, 5, 63bitr4i 268 . 2  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  C.  B
)
81, 7bitri 240 1  |-  ( A  e.  B  <->  ( B  \  { A } ) 
C.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152    C. wpss 3153   {csn 3640
This theorem is referenced by:  mrieqv2d  13541
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-in 3159  df-ss 3166  df-pss 3168  df-sn 3646
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