MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pssn2lp Structured version   Unicode version

Theorem pssn2lp 3434
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 3419 . . . 4  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  B  C_  A ) )
21simprbi 452 . . 3  |-  ( A 
C.  B  ->  -.  B  C_  A )
3 pssss 3428 . . 3  |-  ( B 
C.  A  ->  B  C_  A )
42, 3nsyl 116 . 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 3    -> wi 4    /\ wa 360    C_ wss 3306    C. wpss 3307
This theorem is referenced by:  psstr  3437  cvnsym  23824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-ne 2607  df-in 3313  df-ss 3320  df-pss 3322
  Copyright terms: Public domain W3C validator