Theorem pssn2lp 3045

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

Step	Hyp	Ref	Expression
1		dfpss3 3030	. . . 4 |- ( A C. B <-> ( A C_ B /\ -. B C_ A ) )
2	1	simprbi 260	. . 3 |- ( A C. B -> -. B C_ A )
3		pssss 3039	. . 3 |- ( B C. A -> B C_ A )
4	2, 3	nsyl 558	. 2 |- ( A C. B -> -. B C. A )
5		imnan 624	. 2 |- ( ( A C. B -> -. B C. A ) <-> -. ( A C. B /\ B C. A ) )
6	4, 5	mpbi 133	1 |- -. ( A C. B /\ B C. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   C_ wss 2917   C. wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by:  psstr  3049