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Theorem sspsstr 3454
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
sspsstr  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )

Proof of Theorem sspsstr
StepHypRef Expression
1 sspss 3448 . 2  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
2 psstr 3453 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 425 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq1 3436 . . . . 5  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
54biimprd 216 . . . 4  |-  ( A  =  B  ->  ( B  C.  C  ->  A  C.  C ) )
63, 5jaoi 370 . . 3  |-  ( ( A  C.  B  \/  A  =  B )  ->  ( B  C.  C  ->  A  C.  C ) )
76imp 420 . 2  |-  ( ( ( A  C.  B  \/  A  =  B
)  /\  B  C.  C )  ->  A  C.  C )
81, 7sylanb 460 1  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  sspsstrd  3457  ordtr2  4627  php  7293  canthp1lem2  8530  suplem1pr  8931  fbfinnfr  17875  ppiltx  20962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-in 3329  df-ss 3336  df-pss 3338
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