Theorem sotricim 4253

Description: One direction of sotritric 4254 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 4247	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
2	1	adantrr 471	. . . . . 6 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3	2	3adant3 1002	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B R B )
4		breq2 3941	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	biimprcd 159	. . . . . 6 |- ( B R C -> ( B = C -> B R B ) )
6	5	3ad2ant3 1005	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> ( B = C -> B R B ) )
7	3, 6	mtod 653	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B = C )
8	7	3expia 1184	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. B = C ) )
9		so2nr 4251	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
10		imnan 680	. . . 4 |- ( ( B R C -> -. C R B ) <-> -. ( B R C /\ C R B ) )
11	9, 10	sylibr 133	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. C R B ) )
12	8, 11	jcad 305	. 2 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> ( -. B = C /\ -. C R B ) ) )
13		ioran 742	. 2 |- ( -. ( B = C \/ C R B ) <-> ( -. B = C /\ -. C R B ) )
14	12, 13	syl6ibr 161	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 103   \/ wo 698   /\ w3a 963   = wceq 1332   e. wcel 1481   class class class wbr 3937   Or wor 4225

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-po 4226  df-iso 4227

This theorem is referenced by:  sotritric  4254