Theorem ssnpss 2149

Description: Partial trichotomy law for subclasses.

Assertion

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

Proof of Theorem ssnpss

Step	Hyp	Ref	Expression
1		sspss 2145	. 2 |- (A (_ B <-> (A (. B \/ A = B))
2		pssn2lp 2147	. . . 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 2146	. . . 4 |- -. A (. A
6		psseq1 2135	. . . 4 |- (A = B -> (A (. A <-> B (. A))
7	5, 6	mtbii 716	. . 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 956   (_ wss 2047   (. wpss 2048

This theorem is referenced by:  suplem2pr 5162  atcvat 10313

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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