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Theorem psstr 3293
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 3284 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
2 pssss 3284 . . 3  |-  ( B 
C.  C  ->  B  C_  C )
31, 2sylan9ss 3205 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C_  C )
4 pssn2lp 3290 . . . 4  |-  -.  ( C  C.  B  /\  B  C.  C )
5 psseq1 3276 . . . . 5  |-  ( A  =  C  ->  ( A  C.  B  <->  C  C.  B ) )
65anbi1d 685 . . . 4  |-  ( A  =  C  ->  (
( A  C.  B  /\  B  C.  C )  <-> 
( C  C.  B  /\  B  C.  C ) ) )
74, 6mtbiri 294 . . 3  |-  ( A  =  C  ->  -.  ( A  C.  B  /\  B  C.  C ) )
87con2i 112 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  -.  A  =  C
)
9 dfpss2 3274 . 2  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
103, 8, 9sylanbrc 645 1  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  sspsstr  3294  psssstr  3295  psstrd  3296  porpss  6297  inf3lem5  7349  ltsopr  8672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179  df-pss 3181
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