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Theorem pssn2lp 3544
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 3529 . . . 4  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
21simprbi 462 . . 3  |-  ( A 
C.  B  ->  -.  B  C_  A )
3 pssss 3538 . . 3  |-  ( B 
C.  A  ->  B  C_  A )
42, 3nsyl 121 . 2  |-  ( A 
C.  B  ->  -.  B  C.  A )
5 imnan 420 . 2  |-  ( ( A  C.  B  ->  -.  B  C.  A )  <->  -.  ( A  C.  B  /\  B  C.  A ) )
64, 5mpbi 208 1  |-  -.  ( A  C.  B  /\  B  C.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    C_ wss 3414    C. wpss 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-ne 2600  df-in 3421  df-ss 3428  df-pss 3430
This theorem is referenced by:  psstr  3547  cvnsym  27622
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