Theorem sotrieq 2852

Description: Trichotomy law for strict order relation.

Assertion

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

Proof of Theorem sotrieq

Step	Hyp	Ref	Expression
1		breq2 2613	. . . . . . . 8 |- (B = C -> (BRB <-> BRC))
2	1	negbid 609	. . . . . . 7 |- (B = C -> (-. BRB <-> -. BRC))
3		sonr 2846	. . . . . . 7 |- ((R Or A /\ B e. A) -> -. BRB)
4	2, 3	syl5bi 208	. . . . . 6 |- (B = C -> ((R Or A /\ B e. A) -> -. BRC))
5		breq2 2613	. . . . . . . 8 |- (B = C -> (CRB <-> CRC))
6	5	negbid 609	. . . . . . 7 |- (B = C -> (-. CRB <-> -. CRC))
7		sonr 2846	. . . . . . 7 |- ((R Or A /\ C e. A) -> -. CRC)
8	6, 7	syl5bir 210	. . . . . 6 |- (B = C -> ((R Or A /\ C e. A) -> -. CRB))
9	4, 8	anim12d 556	. . . . 5 |- (B = C -> (((R Or A /\ B e. A) /\ (R Or A /\ C e. A)) -> (-. BRC /\ -. CRB)))
10	9	com12 11	. . . 4 |- (((R Or A /\ B e. A) /\ (R Or A /\ C e. A)) -> (B = C -> (-. BRC /\ -. CRB)))
11	10	anandis 511	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C -> (-. BRC /\ -. CRB)))
12		sotric 2851	. . . . . . . . 9 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC <-> -. (B = C \/ CRB)))
13	12	con2bid 524	. . . . . . . 8 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((B = C \/ CRB) <-> -. BRC))
14	13	biimpar 417	. . . . . . 7 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (B = C \/ CRB))
15	14	ord 232	. . . . . 6 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (-. B = C -> CRB))
16	15	con1d 93	. . . . 5 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (-. CRB -> B = C))
17	16	ex 373	. . . 4 |- ((R Or A /\ (B e. A /\ C e. A)) -> (-. BRC -> (-. CRB -> B = C)))
18	17	imp3a 361	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((-. BRC /\ -. CRB) -> B = C))
19	11, 18	impbid 514	. 2 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> (-. BRC /\ -. CRB)))
20		ioran 306	. 2 |- (-. (BRC \/ CRB) <-> (-. BRC /\ -. CRB))
21	19, 20	syl6bbr 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 953   e. wcel 955   class class class wbr 2609   Or wor 2830

This theorem is referenced by:  sotrieq2 2853  distrlem4pr 5102  addcanpr 5124  sqgt0sr 5187  lttri2t 5485  xrlttri2t 5528  xrltnet 5538

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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