Theorem sotricim 4308

Description: One direction of sotritric 4309 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 4302	. . . . . . 7 |- ( ( R Or A /\ B e. A ) -> -. B R B )
2	1	adantrr 476	. . . . . 6 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3	2	3adant3 1012	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B R B )
4		breq2 3993	. . . . . . 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 1015	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> ( B = C -> B R B ) )
7	3, 6	mtod 658	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B = C )
8	7	3expia 1200	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. B = C ) )
9		so2nr 4306	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
10		imnan 685	. . . 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 747	. 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 703   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989   Or wor 4280

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-po 4281  df-iso 4282

This theorem is referenced by:  sotritric  4309