Theorem sotric 2860

Description: A strict order relation satisfies strict trichotomy.

Assertion

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

Proof of Theorem sotric

Step	Hyp	Ref	Expression
1		breq2 2623	. . . . . . 7 |- (B = C -> (BRB <-> BRC))
2	1	negbid 611	. . . . . 6 |- (B = C -> (-. BRB <-> -. BRC))
3		sonr 2855	. . . . . 6 |- ((R Or A /\ B e. A) -> -. BRB)
4	2, 3	syl5cbi 209	. . . . 5 |- ((R Or A /\ B e. A) -> (B = C -> -. BRC))
5	4	adantrr 395	. . . 4 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C -> -. BRC))
6		so2nr 2858	. . . . . 6 |- ((R Or A /\ (B e. A /\ C e. A)) -> -. (BRC /\ CRB))
7		imnan 242	. . . . . 6 |- ((BRC -> -. CRB) <-> -. (BRC /\ CRB))
8	6, 7	sylibr 200	. . . . 5 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC -> -. CRB))
9	8	con2d 91	. . . 4 |- ((R Or A /\ (B e. A /\ C e. A)) -> (CRB -> -. BRC))
10	5, 9	jaod 424	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((B = C \/ CRB) -> -. BRC))
11		solin 2857	. . . 4 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC \/ B = C \/ CRB))
12		3orass 778	. . . . 5 |- ((BRC \/ B = C \/ CRB) <-> (BRC \/ (B = C \/ CRB)))
13		df-or 224	. . . . 5 |- ((BRC \/ (B = C \/ CRB)) <-> (-. BRC -> (B = C \/ CRB)))
14	12, 13	bitr 173	. . . 4 |- ((BRC \/ B = C \/ CRB) <-> (-. BRC -> (B = C \/ CRB)))
15	11, 14	sylib 198	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> (-. BRC -> (B = C \/ CRB)))
16	10, 15	impbid 516	. 2 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((B = C \/ CRB) <-> -. BRC))
17	16	con2bid 526	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC <-> -. (B = C \/ CRB)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 774   = wceq 956   e. wcel 958   class class class wbr 2619   Or wor 2839

This theorem is referenced by:  sotrieq 2861  indpi 5034  ltsopq 5075  ltrpq 5085  prub 5098  prlem934b 5138  ltapr 5151  suplem2pr 5162  ltsosr 5203  suppsr2 5223  suppsr3 5224  ltsor 5261  pre-axlttri 5287

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

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