Theorem sotricim 4078

Description: One direction of sotritric 4079 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)

Assertion

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

Proof of Theorem sotricim

Step	Hyp	Ref	Expression
1		sonr 4072	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
2	1	adantrr 462	. . . . . 6 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3	2	3adant3 958	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B R B )
4		breq2 3789	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	biimprcd 158	. . . . . 6 |- ( B R C -> ( B = C -> B R B ) )
6	5	3ad2ant3 961	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> ( B = C -> B R B ) )
7	3, 6	mtod 621	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B = C )
8	7	3expia 1140	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. B = C ) )
9		so2nr 4076	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
10		imnan 656	. . . 4 |- ( ( B R C -> -. C R B ) <-> -. ( B R C /\ C R B ) )
11	9, 10	sylibr 132	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. C R B ) )
12	8, 11	jcad 301	. 2 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> ( -. B = C /\ -. C R B ) ) )
13		ioran 701	. 2 |- ( -. ( B = C \/ C R B ) <-> ( -. B = C /\ -. C R B ) )
14	12, 13	syl6ibr 160	1 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. ( B = C \/ C R B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785   Or wor 4050

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-po 4051  df-iso 4052

This theorem is referenced by:  sotritric  4079