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Theorem difsnpss 4159
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 289 . 2  |-  ( A  e.  B  <->  -.  -.  A  e.  B )
2 difss 3617 . . . 4  |-  ( B 
\  { A }
)  C_  B
32biantrur 504 . . 3  |-  ( ( B  \  { A } )  =/=  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
4 difsnb 4158 . . . 4  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
54necon3bbii 2715 . . 3  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  =/=  B
)
6 df-pss 3477 . . 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 367    e. wcel 1823    =/= wne 2649    \ cdif 3458    C_ wss 3461    C. wpss 3462   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-pss 3477  df-sn 4017
This theorem is referenced by:  marypha1lem  7885  infpss  8588  ominf4  8683  mrieqv2d  15131
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