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Theorem psstr 3411
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 3402 . . 3  |-  ( A 
C.  B  ->  A  C_  B )
2 pssss 3402 . . 3  |-  ( B 
C.  C  ->  B  C_  C )
31, 2sylan9ss 3321 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C_  C )
4 pssn2lp 3408 . . . 4  |-  -.  ( C  C.  B  /\  B  C.  C )
5 psseq1 3394 . . . . 5  |-  ( A  =  C  ->  ( A  C.  B  <->  C  C.  B ) )
65anbi1d 686 . . . 4  |-  ( A  =  C  ->  (
( A  C.  B  /\  B  C.  C )  <-> 
( C  C.  B  /\  B  C.  C ) ) )
74, 6mtbiri 295 . . 3  |-  ( A  =  C  ->  -.  ( A  C.  B  /\  B  C.  C ) )
87con2i 114 . 2  |-  ( ( A  C.  B  /\  B  C.  C )  ->  -.  A  =  C
)
9 dfpss2 3392 . 2  |-  ( A 
C.  C  <->  ( A  C_  C  /\  -.  A  =  C ) )
103, 8, 9sylanbrc 646 1  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3280    C. wpss 3281
This theorem is referenced by:  sspsstr  3412  psssstr  3413  psstrd  3414  porpss  6485  inf3lem5  7543  ltsopr  8865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-ne 2569  df-in 3287  df-ss 3294  df-pss 3296
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