Theorem pssn2lp 2709

Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (The proof was 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 2695	. . . 4 |- (A C. B <-> (A C_ B /\ -. B C_ A))
2	1	simprbi 353	. . 3 |- (A C. B -> -. B C_ A)
3		pssss 2705	. . 3 |- (B C. A -> B C_ A)
4	2, 3	nsyl 131	. 2 |- (A C. B -> -. B C. A)
5		imnan 261	. 2 |- ((A C. B -> -. B C. A) <-> -. (A C. B /\ B C. A))
6	4, 5	mpbi 206	1 |- -. (A C. B /\ B C. A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   C_ wss 2593   C. wpss 2594

This theorem is referenced by:  ssnpssOLD 2713  psstr 2714  cvnsym 11862

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-in 2603  df-ss 2605  df-pss 2607

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