Theorem ssnpss 2146

Description: Partial trichotomy law for subclasses.

Assertion

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

Proof of Theorem ssnpss

Step	Hyp	Ref	Expression
1		sspss 2142	. 2 |- (A (_ B <-> (A (. B \/ A = B))
2		pssn2lp 2144	. . . 4 |- -. (A (. B /\ B (. A)
3		imnan 242	. . . 4 |- ((A (. B -> -. B (. A) <-> -. (A (. B /\ B (. A))
4	2, 3	mpbir 190	. . 3 |- (A (. B -> -. B (. A)
5		pssirr 2143	. . . 4 |- -. A (. A
6		psseq1 2132	. . . 4 |- (A = B -> (A (. A <-> B (. A))
7	5, 6	mtbii 715	. . 3 |- (A = B -> -. B (. A)
8	4, 7	jaoi 341	. 2 |- ((A (. B \/ A = B) -> -. B (. A)
9	1, 8	sylbi 199	1 |- (A (_ B -> -. B (. A)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 955   (_ wss 2044   (. wpss 2045

This theorem is referenced by:  suplem2pr 5145  atcvat 10269

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-in 2048  df-ss 2050  df-pss 2052

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