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Theorem pssn2lp 3197
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 3183 . . . 4  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
21simprbi 452 . . 3  |-  ( A 
C.  B  ->  -.  B  C_  A )
3 pssss 3192 . . 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 3078    C. wpss 3079
This theorem is referenced by:  psstr  3200  cvnsym  22700
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-ne 2414  df-in 3085  df-ss 3089  df-pss 3091
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