Theorem sotritrieq 4062

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 4054	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
3	1, 2	mpan 400	. . . . . 6 |- ( B e. A -> -. B R B )
4		breq2 3768	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	notbid 592	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. B R C ) )
6	3, 5	syl5ibcom 144	. . . . 5 |- ( B e. A -> ( B = C -> -. B R C ) )
7		breq1 3767	. . . . . . 7 |- ( B = C -> ( B R B <-> C R B ) )
8	7	notbid 592	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. C R B ) )
9	3, 8	syl5ibcom 144	. . . . 5 |- ( B e. A -> ( B = C -> -. C R B ) )
10	6, 9	jcad 291	. . . 4 |- ( B e. A -> ( B = C -> ( -. B R C /\ -. C R B ) ) )
11		ioran 669	. . . 4 |- ( -. ( B R C \/ C R B ) <-> ( -. B R C /\ -. C R B ) )
12	10, 11	syl6ibr 151	. . 3 |- ( B e. A -> ( B = C -> -. ( B R C \/ C R B ) ) )
13	12	adantr 261	. 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 891	. . . . . . 7 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ C R B \/ B R C ) )
16		3orcomb 894	. . . . . . 7 |- ( ( B = C \/ C R B \/ B R C ) <-> ( B = C \/ B R C \/ C R B ) )
17		3orass 888	. . . . . . 7 |- ( ( B = C \/ B R C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
18	15, 16, 17	3bitri 195	. . . . . 6 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
19	18	biimpi 113	. . . . 5 |- ( ( B R C \/ B = C \/ C R B ) -> ( B = C \/ ( B R C \/ C R B ) ) )
20	19	orcomd 648	. . . 4 |- ( ( B R C \/ B = C \/ C R B ) -> ( ( B R C \/ C R B ) \/ B = C ) )
21	20	ord 643	. . 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 120	1 |- ( ( B e. A /\ C e. A ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   \/ w3o 884   = wceq 1243   e. wcel 1393   class class class wbr 3764   Or wor 4032

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034

This theorem is referenced by:  distrlem4prl  6682  distrlem4pru  6683