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Theorem sspss 3288
Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
Assertion
Ref Expression
sspss  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )

Proof of Theorem sspss
StepHypRef Expression
1 dfpss2 3274 . . . . 5  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
21simplbi2 608 . . . 4  |-  ( A 
C_  B  ->  ( -.  A  =  B  ->  A  C.  B ) )
32con1d 116 . . 3  |-  ( A 
C_  B  ->  ( -.  A  C.  B  ->  A  =  B )
)
43orrd 367 . 2  |-  ( A 
C_  B  ->  ( A  C.  B  \/  A  =  B ) )
5 pssss 3284 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
6 eqimss 3243 . . 3  |-  ( A  =  B  ->  A  C_  B )
75, 6jaoi 368 . 2  |-  ( ( A  C.  B  \/  A  =  B )  ->  A  C_  B )
84, 7impbii 180 1  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    = wceq 1632    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  sspsstri  3291  sspsstr  3294  psssstr  3295  ordsseleq  4437  sorpssuni  6302  sorpssint  6303  ssnnfi  7098  ackbij1b  7881  fin23lem40  7993  zorng  8147  psslinpr  8671  suplem2pr  8693  mrissmrcd  13558  pgpssslw  14941  pgpfac1lem5  15330  idnghm  18268  dfon2lem4  24213  finminlem  26334  lkrss2N  29981  dvh3dim3N  32261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179  df-pss 3181
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