Theorem sotrieq2 2853

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 2852	. 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 534	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 953   e. wcel 955   class class class wbr 2609   Or wor 2830

This theorem is referenced by:  supmo 4550  lttri3t 5486  xrlttri3t 5529

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-po 2831  df-so 2841

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