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Theorem sspss 3438
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 3424 . . . . 5  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
21simplbi2 609 . . . 4  |-  ( A 
C_  B  ->  ( -.  A  =  B  ->  A  C.  B ) )
32con1d 118 . . 3  |-  ( A 
C_  B  ->  ( -.  A  C.  B  ->  A  =  B )
)
43orrd 368 . 2  |-  ( A 
C_  B  ->  ( A  C.  B  \/  A  =  B ) )
5 pssss 3434 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
6 eqimss 3392 . . 3  |-  ( A  =  B  ->  A  C_  B )
75, 6jaoi 369 . 2  |-  ( ( A  C.  B  \/  A  =  B )  ->  A  C_  B )
84, 7impbii 181 1  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    = wceq 1652    C_ wss 3312    C. wpss 3313
This theorem is referenced by:  sspsstri  3441  sspsstr  3444  psssstr  3445  ordsseleq  4602  sorpssuni  6523  sorpssint  6524  ssnnfi  7320  ackbij1b  8111  fin23lem40  8223  zorng  8376  psslinpr  8900  suplem2pr  8922  mrissmrcd  13857  pgpssslw  15240  pgpfac1lem5  15629  idnghm  18769  dfon2lem4  25405  finminlem  26312  lkrss2N  29904  dvh3dim3N  32184
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-ne 2600  df-in 3319  df-ss 3326  df-pss 3328
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