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Theorem psstr 3455
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 3446 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
2 pssss 3446 . . 3  |-  ( B 
C.  C  ->  B  C_  C )
31, 2sylan9ss 3364 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C_  C )
4 pssn2lp 3452 . . . 4  |-  -.  ( C  C.  B  /\  B  C.  C )
5 psseq1 3438 . . . . 5  |-  ( A  =  C  ->  ( A  C.  B  <->  C  C.  B
) )
65anbi1d 704 . . . 4  |-  ( A  =  C  ->  (
( A  C.  B  /\  B  C.  C )  <-> 
( C  C.  B  /\  B  C.  C ) ) )
74, 6mtbiri 303 . . 3  |-  ( A  =  C  ->  -.  ( A  C.  B  /\  B  C.  C ) )
87con2i 120 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  -.  A  =  C
)
9 dfpss2 3436 . 2  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
103, 8, 9sylanbrc 664 1  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    C_ wss 3323    C. wpss 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-in 3330  df-ss 3337  df-pss 3339
This theorem is referenced by:  sspsstr  3456  psssstr  3457  psstrd  3458  porpss  6359  inf3lem5  7830  ltsopr  9193
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