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Theorem difsnpss 4111
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 291 . 2  |-  ( A  e.  B  <->  -.  -.  A  e.  B )
2 difss 3578 . . . 4  |-  ( B 
\  { A }
)  C_  B
32biantrur 506 . . 3  |-  ( ( B  \  { A } )  =/=  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
4 difsnb 4110 . . . 4  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
54necon3bbii 2707 . . 3  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  =/=  B
)
6 df-pss 3439 . . 3  |-  ( ( B  \  { A } )  C.  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
73, 5, 63bitr4i 277 . 2  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  C.  B
)
81, 7bitri 249 1  |-  ( A  e.  B  <->  ( B  \  { A } ) 
C.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2642    \ cdif 3420    C_ wss 3423    C. wpss 3424   {csn 3972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-v 3067  df-dif 3426  df-in 3430  df-ss 3437  df-pss 3439  df-sn 3973
This theorem is referenced by:  marypha1lem  7781  infpss  8484  ominf4  8579  mrieqv2d  14676
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