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Theorem pssn2lp 3278
Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
pssn2lp  |-  -.  ( A  C.  B  /\  B  C.  A )

Proof of Theorem pssn2lp
StepHypRef Expression
1 dfpss3 3263 . . . 4  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
21simprbi 452 . . 3  |-  ( A 
C.  B  ->  -.  B  C_  A )
3 pssss 3272 . . 3  |-  ( B 
C.  A  ->  B  C_  A )
42, 3nsyl 115 . 2  |-  ( A 
C.  B  ->  -.  B  C.  A )
5 imnan 413 . 2  |-  ( ( A  C.  B  ->  -.  B  C.  A )  <->  -.  ( A  C.  B  /\  B  C.  A ) )
64, 5mpbi 201 1  |-  -.  ( A  C.  B  /\  B  C.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    C_ wss 3153    C. wpss 3154
This theorem is referenced by:  psstr  3281  cvnsym  22862
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-ne 2449  df-in 3160  df-ss 3167  df-pss 3169
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