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Theorem psstr 3280
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 3271 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
2 pssss 3271 . . 3  |-  ( B 
C.  C  ->  B  C_  C )
31, 2sylan9ss 3192 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C_  C )
4 pssn2lp 3277 . . . 4  |-  -.  ( C  C.  B  /\  B  C.  C )
5 psseq1 3263 . . . . 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 3261 . 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 1623    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  sspsstr  3281  psssstr  3282  psstrd  3283  porpss  6281  inf3lem5  7333  ltsopr  8656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-in 3159  df-ss 3166  df-pss 3168
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