Theorem pssn2lp 1571

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		pm3.24 496	. 2 |- -. ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A))
2		dfpss3 1558	. . . . 5 |- (A (. B <-> (A (_ B /\ -. B (_ A))
3		dfpss3 1558	. . . . 5 |- (B (. A <-> (B (_ A /\ -. A (_ B))
4	2, 3	anbi12i 369	. . . 4 |- ((A (. B /\ B (. A) <-> ((A (_ B /\ -. B (_ A) /\ (B (_ A /\ -. A (_ B)))
5		an42 389	. . . 4 |- (((A (_ B /\ -. B (_ A) /\ (B (_ A /\ -. A (_ B)) <-> ((A (_ B /\ B (_ A) /\ (-. A (_ B /\ -. B (_ A)))
6	4, 5	bitr 151	. . 3 |- ((A (. B /\ B (. A) <-> ((A (_ B /\ B (_ A) /\ (-. A (_ B /\ -. B (_ A)))
7		orc 225	. . . . . 6 |- (-. A (_ B -> (-. A (_ B \/ -. B (_ A))
8	7	adantr 306	. . . . 5 |- ((-. A (_ B /\ -. B (_ A) -> (-. A (_ B \/ -. B (_ A))
9		ianor 253	. . . . 5 |- (-. (A (_ B /\ B (_ A) <-> (-. A (_ B \/ -. B (_ A))
10	8, 9	sylibr 175	. . . 4 |- ((-. A (_ B /\ -. B (_ A) -> -. (A (_ B /\ B (_ A))
11	10	anim2i 270	. . 3 |- (((A (_ B /\ B (_ A) /\ (-. A (_ B /\ -. B (_ A)) -> ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A)))
12	6, 11	sylbi 174	. 2 |- ((A (. B /\ B (. A) -> ((A (_ B /\ B (_ A) /\ -. (A (_ B /\ B (_ A)))
13	1, 12	mto 93	1 |- -. (A (. B /\ B (. A)

Syntax hints:  -. wn 1   \/ wo 195   /\ wa 196   (_ wss 1487   (. wpss 1488

This theorem is referenced by:  ssnpss 1573  psstr 1574  cvnsymt 5722

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-in 1491  df-ss 1492  df-pss 1494

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