MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difsnpss Structured version   Unicode version

Theorem difsnpss 3933
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 3466 . . . 4  |-  ( B 
\  { A }
)  C_  B
32biantrur 493 . . 3  |-  ( ( B  \  { A } )  =/=  B  <->  ( ( B  \  { A } )  C_  B  /\  ( B  \  { A } )  =/=  B
) )
4 difsnb 3932 . . . 4  |-  ( -.  A  e.  B  <->  ( B  \  { A } )  =  B )
54necon3bbii 2629 . . 3  |-  ( -. 
-.  A  e.  B  <->  ( B  \  { A } )  =/=  B
)
6 df-pss 3328 . . 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 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312    C. wpss 3313   {csn 3806
This theorem is referenced by:  marypha1lem  7430  infpss  8089  ominf4  8184  mrieqv2d  13856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-pss 3328  df-sn 3812
  Copyright terms: Public domain W3C validator