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Theorem sspss 3476
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 3462 . . . . 5  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
21simplbi2 625 . . . 4  |-  ( A 
C_  B  ->  ( -.  A  =  B  ->  A  C.  B )
)
32con1d 124 . . 3  |-  ( A 
C_  B  ->  ( -.  A  C.  B  ->  A  =  B )
)
43orrd 378 . 2  |-  ( A 
C_  B  ->  ( A  C.  B  \/  A  =  B ) )
5 pssss 3472 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
6 eqimss 3429 . . 3  |-  ( A  =  B  ->  A  C_  B )
75, 6jaoi 379 . 2  |-  ( ( A  C.  B  \/  A  =  B )  ->  A  C_  B )
84, 7impbii 188 1  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    = wceq 1369    C_ wss 3349    C. wpss 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-ne 2622  df-in 3356  df-ss 3363  df-pss 3365
This theorem is referenced by:  sspsstri  3479  sspsstr  3482  psssstr  3483  ordsseleq  4769  sorpssuni  6390  sorpssint  6391  ssnnfi  7553  ackbij1b  8429  fin23lem40  8541  zorng  8694  psslinpr  9221  suplem2pr  9243  mrissmrcd  14599  pgpssslw  16134  pgpfac1lem5  16602  idnghm  20344  dfon2lem4  27621  finminlem  28539  lkrss2N  32910  dvh3dim3N  35190
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