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Theorem sspsstr 3570
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 3564 . 2  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
2 psstr 3569 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 434 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq1 3552 . . . . 5  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C
) )
54biimprd 223 . . . 4  |-  ( A  =  B  ->  ( B  C.  C  ->  A  C.  C ) )
63, 5jaoi 379 . . 3  |-  ( ( A  C.  B  \/  A  =  B )  ->  ( B  C.  C  ->  A  C.  C )
)
76imp 429 . 2  |-  ( ( ( A  C.  B  \/  A  =  B
)  /\  B  C.  C
)  ->  A  C.  C
)
81, 7sylanb 472 1  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    C_ wss 3437    C. wpss 3438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-ne 2650  df-in 3444  df-ss 3451  df-pss 3453
This theorem is referenced by:  sspsstrd  3573  ordtr2  4872  php  7606  canthp1lem2  8932  suplem1pr  9333  fbfinnfr  19547  ppiltx  22649
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