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Theorem psssstr 3596
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
psssstr  |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )

Proof of Theorem psssstr
StepHypRef Expression
1 sspss 3589 . 2  |-  ( B 
C_  C  <->  ( B  C.  C  \/  B  =  C ) )
2 psstr 3594 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 432 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq2 3578 . . . . 5  |-  ( B  =  C  ->  ( A  C.  B  <->  A  C.  C
) )
54biimpcd 224 . . . 4  |-  ( A 
C.  B  ->  ( B  =  C  ->  A 
C.  C ) )
63, 5jaod 378 . . 3  |-  ( A 
C.  B  ->  (
( B  C.  C  \/  B  =  C
)  ->  A  C.  C
) )
76imp 427 . 2  |-  ( ( A  C.  B  /\  ( B  C.  C  \/  B  =  C )
)  ->  A  C.  C
)
81, 7sylan2b 473 1  |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    C_ wss 3461    C. wpss 3462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ne 2651  df-in 3468  df-ss 3475  df-pss 3477
This theorem is referenced by:  psssstrd  3599  suplem1pr  9419  atexch  27498  bj-2upln0  34982  bj-2upln1upl  34983
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