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Theorem pssn2lp 3600
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 3585 . . . 4  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
21simprbi 464 . . 3  |-  ( A 
C.  B  ->  -.  B  C_  A )
3 pssss 3594 . . 3  |-  ( B 
C.  A  ->  B  C_  A )
42, 3nsyl 121 . 2  |-  ( A 
C.  B  ->  -.  B  C.  A )
5 imnan 422 . 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 369    C_ wss 3471    C. wpss 3472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-ne 2659  df-in 3478  df-ss 3485  df-pss 3487
This theorem is referenced by:  psstr  3603  cvnsym  26873
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