Theorem sotritrieq 4161

Description: A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)

Hypotheses

Ref	Expression
sotritric.or	|- R Or A
sotritric.tri	|- ( ( B e. A /\ C e. A ) -> ( B R C \/ B = C \/ C R B ) )

Assertion

Ref	Expression
sotritrieq	|- ( ( B e. A /\ C e. A ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Proof of Theorem sotritrieq

Step	Hyp	Ref	Expression
1		sotritric.or	. . . . . . 7 |- R Or A
2		sonr 4153	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
3	1, 2	mpan 416	. . . . . 6 |- ( B e. A -> -. B R B )
4		breq2 3855	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	notbid 628	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. B R C ) )
6	3, 5	syl5ibcom 154	. . . . 5 |- ( B e. A -> ( B = C -> -. B R C ) )
7		breq1 3854	. . . . . . 7 |- ( B = C -> ( B R B <-> C R B ) )
8	7	notbid 628	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. C R B ) )
9	3, 8	syl5ibcom 154	. . . . 5 |- ( B e. A -> ( B = C -> -. C R B ) )
10	6, 9	jcad 302	. . . 4 |- ( B e. A -> ( B = C -> ( -. B R C /\ -. C R B ) ) )
11		ioran 705	. . . 4 |- ( -. ( B R C \/ C R B ) <-> ( -. B R C /\ -. C R B ) )
12	10, 11	syl6ibr 161	. . 3 |- ( B e. A -> ( B = C -> -. ( B R C \/ C R B ) ) )
13	12	adantr 271	. 2 |- ( ( B e. A /\ C e. A ) -> ( B = C -> -. ( B R C \/ C R B ) ) )
14		sotritric.tri	. . 3 |- ( ( B e. A /\ C e. A ) -> ( B R C \/ B = C \/ C R B ) )
15		3orrot 931	. . . . . . 7 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ C R B \/ B R C ) )
16		3orcomb 934	. . . . . . 7 |- ( ( B = C \/ C R B \/ B R C ) <-> ( B = C \/ B R C \/ C R B ) )
17		3orass 928	. . . . . . 7 |- ( ( B = C \/ B R C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
18	15, 16, 17	3bitri 205	. . . . . 6 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
19	18	biimpi 119	. . . . 5 |- ( ( B R C \/ B = C \/ C R B ) -> ( B = C \/ ( B R C \/ C R B ) ) )
20	19	orcomd 684	. . . 4 |- ( ( B R C \/ B = C \/ C R B ) -> ( ( B R C \/ C R B ) \/ B = C ) )
21	20	ord 679	. . 3 |- ( ( B R C \/ B = C \/ C R B ) -> ( -. ( B R C \/ C R B ) -> B = C ) )
22	14, 21	syl 14	. 2 |- ( ( B e. A /\ C e. A ) -> ( -. ( B R C \/ C R B ) -> B = C ) )
23	13, 22	impbid 128	1 |- ( ( B e. A /\ C e. A ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 665   \/ w3o 924   = wceq 1290   e. wcel 1439   class class class wbr 3851   Or wor 4131

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-po 4132  df-iso 4133

This theorem is referenced by:  distrlem4prl  7204  distrlem4pru  7205