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Theorem psstr 3453
Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psstr  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )

Proof of Theorem psstr
StepHypRef Expression
1 pssss 3444 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
2 pssss 3444 . . 3  |-  ( B 
C.  C  ->  B  C_  C )
31, 2sylan9ss 3363 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C_  C )
4 pssn2lp 3450 . . . 4  |-  -.  ( C  C.  B  /\  B  C.  C )
5 psseq1 3436 . . . . 5  |-  ( A  =  C  ->  ( A  C.  B  <->  C  C.  B ) )
65anbi1d 687 . . . 4  |-  ( A  =  C  ->  (
( A  C.  B  /\  B  C.  C )  <-> 
( C  C.  B  /\  B  C.  C ) ) )
74, 6mtbiri 296 . . 3  |-  ( A  =  C  ->  -.  ( A  C.  B  /\  B  C.  C ) )
87con2i 115 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  -.  A  =  C
)
9 dfpss2 3434 . 2  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
103, 8, 9sylanbrc 647 1  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  sspsstr  3454  psssstr  3455  psstrd  3456  porpss  6528  inf3lem5  7589  ltsopr  8911
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-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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