Theorem pssn2lp 2147

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		pm3.24 658	. 2 |- -. ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A))
2		dfpss3 2134	. . . . 5 |- (A (. B <-> (A (_ B /\ -. B (_ A))
3		dfpss3 2134	. . . . 5 |- (B (. A <-> (B (_ A /\ -. A (_ B))
4	2, 3	anbi12i 482	. . . 4 |- ((A (. B /\ B (. A) <-> ((A (_ B /\ -. B (_ A) /\ (B (_ A /\ -. A (_ B)))
5		an42 507	. . . 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 2047   (. wpss 2048

This theorem is referenced by:  ssnpss 2149  psstr 2150  cvnsymt 10217

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-in 2051  df-ss 2053  df-pss 2055

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