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Theorem psssstr 3563
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 3556 . 2  |-  ( B 
C_  C  <->  ( B  C.  C  \/  B  =  C ) )
2 psstr 3561 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 434 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq2 3545 . . . . 5  |-  ( B  =  C  ->  ( A  C.  B  <->  A  C.  C
) )
54biimpcd 224 . . . 4  |-  ( A 
C.  B  ->  ( B  =  C  ->  A 
C.  C ) )
63, 5jaod 380 . . 3  |-  ( A 
C.  B  ->  (
( B  C.  C  \/  B  =  C
)  ->  A  C.  C
) )
76imp 429 . 2  |-  ( ( A  C.  B  /\  ( B  C.  C  \/  B  =  C )
)  ->  A  C.  C
)
81, 7sylan2b 475 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 3429    C. wpss 3430
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 1952  ax-ext 2430
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 2437  df-cleq 2443  df-clel 2446  df-ne 2646  df-in 3436  df-ss 3443  df-pss 3445
This theorem is referenced by:  psssstrd  3566  suplem1pr  9325  atexch  25930  bj-2upln0  32819  bj-2upln1upl  32820
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