Theorem ssnpss 1573

Description: Partial trichotomy law for subclasses.

Assertion

Ref	Expression
ssnpss	|- (A (_ B -> -. B (. A)

Proof of Theorem ssnpss

Step	Hyp	Ref	Expression
1		sspss 1569	. 2 |- (A (_ B <-> (A (. B \/ A = B))
2		pssn2lp 1571	. . . 4 |- -. (A (. B /\ B (. A)
3		imnan 207	. . . 4 |- ((A (. B -> -. B (. A) <-> -. (A (. B /\ B (. A))
4	2, 3	mpbir 165	. . 3 |- (A (. B -> -. B (. A)
5		pssirr 1570	. . . 4 |- -. A (. A
6		psseq1 1559	. . . 4 |- (A = B -> (A (. A <-> B (. A))
7	5, 6	mtbii 538	. . 3 |- (A = B -> -. B (. A)
8	4, 7	jaoi 275	. 2 |- ((A (. B \/ A = B) -> -. B (. A)
9	1, 8	sylbi 174	1 |- (A (_ B -> -. B (. A)

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   = wceq 1091   (_ wss 1487   (. wpss 1488

This theorem is referenced by:  suplem2pr 3956  atcvat 5771

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

metamath.org