Theorem pssn2lp 2137

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

Assertion

Ref	Expression
pssn2lp	|- -. (A (. B /\ B (. A)

Proof of Theorem pssn2lp

Step	Hyp	Ref	Expression
1		pm3.24 656	. 2 |- -. ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A))
2		dfpss3 2124	. . . . 5 |- (A (. B <-> (A (_ B /\ -. B (_ A))
3		dfpss3 2124	. . . . 5 |- (B (. A <-> (B (_ A /\ -. A (_ B))
4	2, 3	anbi12i 481	. . . 4 |- ((A (. B /\ B (. A) <-> ((A (_ B /\ -. B (_ A) /\ (B (_ A /\ -. A (_ B)))
5		an42 506	. . . 4 |- (((A (_ B /\ -. B (_ A) /\ (B (_ A /\ -. A (_ B)) <-> ((A (_ B /\ B (_ A) /\ (-. A (_ B /\ -. B (_ A)))
6	4, 5	bitr 173	. . 3 |- ((A (. B /\ B (. A) <-> ((A (_ B /\ B (_ A) /\ (-. A (_ B /\ -. B (_ A)))
7		orc 269	. . . . . 6 |- (-. A (_ B -> (-. A (_ B \/ -. B (_ A))
8	7	adantr 389	. . . . 5 |- ((-. A (_ B /\ -. B (_ A) -> (-. A (_ B \/ -. B (_ A))
9		ianor 305	. . . . 5 |- (-. (A (_ B /\ B (_ A) <-> (-. A (_ B \/ -. B (_ A))
10	8, 9	sylibr 200	. . . 4 |- ((-. A (_ B /\ -. B (_ A) -> -. (A (_ B /\ B (_ A))
11	10	anim2i 335	. . 3 |- (((A (_ B /\ B (_ A) /\ (-. A (_ B /\ -. B (_ A)) -> ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A)))
12	6, 11	sylbi 199	. 2 |- ((A (. B /\ B (. A) -> ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A)))
13	1, 12	mto 106	1 |- -. (A (. B /\ B (. A)

Syntax hints:  -. wn 2   \/ wo 222   /\ wa 223   (_ wss 2037   (. wpss 2038

This theorem is referenced by:  ssnpss 2139  psstr 2140  cvnsymt 10127

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-in 2041  df-ss 2043  df-pss 2045

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