Theorem pssn2lpOLD 2710

Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23.

Assertion

Ref	Expression
pssn2lpOLD	|- -. (A C. B /\ B C. A)

Proof of Theorem pssn2lpOLD

Step	Hyp	Ref	Expression
1		pm3.24 720	. 2 |- -. ((A C_ B /\ B C_ A) /\ -. (A C_ B /\ B C_ A))
2		dfpss3 2695	. . . . 5 |- (A C. B <-> (A C_ B /\ -. B C_ A))
3		dfpss3 2695	. . . . 5 |- (B C. A <-> (B C_ A /\ -. A C_ B))
4	2, 3	anbi12i 540	. . . 4 |- ((A C. B /\ B C. A) <-> ((A C_ B /\ -. B C_ A) /\ (B C_ A /\ -. A C_ B)))
5		an42 565	. . . 4 |- (((A C_ B /\ -. B C_ A) /\ (B C_ A /\ -. A C_ B)) <-> ((A C_ B /\ B C_ A) /\ (-. A C_ B /\ -. B C_ A)))
6	4, 5	bitri 190	. . 3 |- ((A C. B /\ B C. A) <-> ((A C_ B /\ B C_ A) /\ (-. A C_ B /\ -. B C_ A)))
7		orc 291	. . . . . 6 |- (-. A C_ B -> (-. A C_ B \/ -. B C_ A))
8	7	adantr 425	. . . . 5 |- ((-. A C_ B /\ -. B C_ A) -> (-. A C_ B \/ -. B C_ A))
9		ianor 329	. . . . 5 |- (-. (A C_ B /\ B C_ A) <-> (-. A C_ B \/ -. B C_ A))
10	8, 9	sylibr 217	. . . 4 |- ((-. A C_ B /\ -. B C_ A) -> -. (A C_ B /\ B C_ A))
11	10	anim2i 362	. . 3 |- (((A C_ B /\ B C_ A) /\ (-. A C_ B /\ -. B C_ A)) -> ((A C_ B /\ B C_ A) /\ -. (A C_ B /\ B C_ A)))
12	6, 11	sylbi 216	. 2 |- ((A C. B /\ B C. A) -> ((A C_ B /\ B C_ A) /\ -. (A C_ B /\ B C_ A)))
13	1, 12	mto 121	1 |- -. (A C. B /\ B C. A)

Syntax hints:  -. wn 2   \/ wo 239   /\ wa 240   C_ wss 2593   C. wpss 2594

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-or 241  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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