Theorem sotrieq2 2862

Description: Trichotomy law for strict order relation.

Assertion

Ref	Expression
sotrieq2	|- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> (-. BRC /\ -. CRB)))

Proof of Theorem sotrieq2

Step	Hyp	Ref	Expression
1		sotrieq 2861	. 2 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> -. (BRC \/ CRB)))
2		ioran 306	. 2 |- (-. (BRC \/ CRB) <-> (-. BRC /\ -. CRB))
3	1, 2	syl6bb 536	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> (-. BRC /\ -. CRB)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958   class class class wbr 2619   Or wor 2839

This theorem is referenced by:  supmo 4576  supmax 4589  lttri3t 5514  xrlttri3t 5556

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-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-po 2840  df-so 2850

Copyright terms: Public domain