Theorem sotricim 4341

Description: One direction of sotritric 4342 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 4335	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
2	1	adantrr 479	. . . . . 6 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3	2	3adant3 1019	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B R B )
4		breq2 4022	. . . . . . 7 |- ( B = C -> ( B R B <-> B R C ) )
5	4	biimprcd 160	. . . . . 6 |- ( B R C -> ( B = C -> B R B ) )
6	5	3ad2ant3 1022	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> ( B = C -> B R B ) )
7	3, 6	mtod 664	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B = C )
8	7	3expia 1207	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. B = C ) )
9		so2nr 4339	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
10		imnan 691	. . . 4 |- ( ( B R C -> -. C R B ) <-> -. ( B R C /\ C R B ) )
11	9, 10	sylibr 134	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. C R B ) )
12	8, 11	jcad 307	. 2 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> ( -. B = C /\ -. C R B ) ) )
13		ioran 753	. 2 |- ( -. ( B = C \/ C R B ) <-> ( -. B = C /\ -. C R B ) )
14	12, 13	imbitrrdi 162	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 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4018   Or wor 4313

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-iso 4315

This theorem is referenced by:  sotritric  4342