Theorem sotritrieq 4247

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 4239	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
3	1, 2	mpan 420	. . . . . 6 |- ( B e. A -> -. B R B )
4		breq2 3933	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	notbid 656	. . . . . 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 3932	. . . . . . 7 |- ( B = C -> ( B R B <-> C R B ) )
8	7	notbid 656	. . . . . 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 305	. . . 4 |- ( B e. A -> ( B = C -> ( -. B R C /\ -. C R B ) ) )
11		ioran 741	. . . 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 274	. 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 968	. . . . . . 7 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ C R B \/ B R C ) )
16		3orcomb 971	. . . . . . 7 |- ( ( B = C \/ C R B \/ B R C ) <-> ( B = C \/ B R C \/ C R B ) )
17		3orass 965	. . . . . . 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 718	. . . 4 |- ( ( B R C \/ B = C \/ C R B ) -> ( ( B R C \/ C R B ) \/ B = C ) )
21	20	ord 713	. . 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 697   \/ w3o 961   = wceq 1331   e. wcel 1480   class class class wbr 3929   Or wor 4217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218  df-iso 4219

This theorem is referenced by:  distrlem4prl  7392  distrlem4pru  7393