Theorem sotritrieq 4152

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 4144	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
3	1, 2	mpan 415	. . . . . 6 |- ( B e. A -> -. B R B )
4		breq2 3849	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	notbid 627	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. B R C ) )
6	3, 5	syl5ibcom 153	. . . . 5 |- ( B e. A -> ( B = C -> -. B R C ) )
7		breq1 3848	. . . . . . 7 |- ( B = C -> ( B R B <-> C R B ) )
8	7	notbid 627	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. C R B ) )
9	3, 8	syl5ibcom 153	. . . . 5 |- ( B e. A -> ( B = C -> -. C R B ) )
10	6, 9	jcad 301	. . . 4 |- ( B e. A -> ( B = C -> ( -. B R C /\ -. C R B ) ) )
11		ioran 704	. . . 4 |- ( -. ( B R C \/ C R B ) <-> ( -. B R C /\ -. C R B ) )
12	10, 11	syl6ibr 160	. . 3 |- ( B e. A -> ( B = C -> -. ( B R C \/ C R B ) ) )
13	12	adantr 270	. 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 930	. . . . . . 7 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ C R B \/ B R C ) )
16		3orcomb 933	. . . . . . 7 |- ( ( B = C \/ C R B \/ B R C ) <-> ( B = C \/ B R C \/ C R B ) )
17		3orass 927	. . . . . . 7 |- ( ( B = C \/ B R C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
18	15, 16, 17	3bitri 204	. . . . . 6 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
19	18	biimpi 118	. . . . 5 |- ( ( B R C \/ B = C \/ C R B ) -> ( B = C \/ ( B R C \/ C R B ) ) )
20	19	orcomd 683	. . . 4 |- ( ( B R C \/ B = C \/ C R B ) -> ( ( B R C \/ C R B ) \/ B = C ) )
21	20	ord 678	. . 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 127	1 |- ( ( B e. A /\ C e. A ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664   \/ w3o 923   = wceq 1289   e. wcel 1438   class class class wbr 3845   Or wor 4122

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-po 4123  df-iso 4124

This theorem is referenced by:  distrlem4prl  7141  distrlem4pru  7142