Theorem sotritrieq 4465

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 4457	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
3	1, 2	mpan 428	. . . . . 6 |- ( B e. A -> -. B R B )
4		breq2 4129	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	notbid 677	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. B R C ) )
6	3, 5	syl5ibcom 155	. . . . 5 |- ( B e. A -> ( B = C -> -. B R C ) )
7		breq1 4128	. . . . . . 7 |- ( B = C -> ( B R B <-> C R B ) )
8	7	notbid 677	. . . . . 6 |- ( B = C -> ( -. B R B <-> -. C R B ) )
9	3, 8	syl5ibcom 155	. . . . 5 |- ( B e. A -> ( B = C -> -. C R B ) )
10	6, 9	jcad 307	. . . 4 |- ( B e. A -> ( B = C -> ( -. B R C /\ -. C R B ) ) )
11		ioran 764	. . . 4 |- ( -. ( B R C \/ C R B ) <-> ( -. B R C /\ -. C R B ) )
12	10, 11	imbitrrdi 162	. . 3 |- ( B e. A -> ( B = C -> -. ( B R C \/ C R B ) ) )
13	12	adantr 276	. 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 1015	. . . . . . 7 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ C R B \/ B R C ) )
16		3orcomb 1018	. . . . . . 7 |- ( ( B = C \/ C R B \/ B R C ) <-> ( B = C \/ B R C \/ C R B ) )
17		3orass 1012	. . . . . . 7 |- ( ( B = C \/ B R C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
18	15, 16, 17	3bitri 206	. . . . . 6 |- ( ( B R C \/ B = C \/ C R B ) <-> ( B = C \/ ( B R C \/ C R B ) ) )
19	18	biimpi 120	. . . . 5 |- ( ( B R C \/ B = C \/ C R B ) -> ( B = C \/ ( B R C \/ C R B ) ) )
20	19	orcomd 741	. . . 4 |- ( ( B R C \/ B = C \/ C R B ) -> ( ( B R C \/ C R B ) \/ B = C ) )
21	20	ord 736	. . 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 129	1 |- ( ( B e. A /\ C e. A ) -> ( B = C <-> -. ( B R C \/ C R B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   \/ w3o 1008   = wceq 1402   e. wcel 2209   class class class wbr 4125   Or wor 4435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-po 4436  df-iso 4437

This theorem is referenced by:  distrlem4prl  7941  distrlem4pru  7942