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Theorem difsnpss 3886
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 283 . 2  |-  ( A  e.  B  <->  -.  -.  A  e.  B )
2 difss 3419 . . . 4  |-  ( B 
\  { A }
)  C_  B
32biantrur 493 . . 3  |-  ( ( B  \  { A } )  =/=  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
4 difsnb 3885 . . . 4  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
54necon3bbii 2583 . . 3  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  =/=  B
)
6 df-pss 3281 . . 3  |-  ( ( B  \  { A } )  C.  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
73, 5, 63bitr4i 269 . 2  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  C.  B
)
81, 7bitri 241 1  |-  ( A  e.  B  <->  ( B  \  { A } ) 
C.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    e. wcel 1717    =/= wne 2552    \ cdif 3262    C_ wss 3265    C. wpss 3266   {csn 3759
This theorem is referenced by:  marypha1lem  7375  infpss  8032  ominf4  8127  mrieqv2d  13793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-v 2903  df-dif 3268  df-in 3272  df-ss 3279  df-pss 3281  df-sn 3765
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