Theorem sotricim 4370

Description: One direction of sotritric 4371 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 4364	. . . . . . 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 1020	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B R B )
4		breq2 4048	. . . . . . 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 1023	. . . . 5 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> ( B = C -> B R B ) )
7	3, 6	mtod 665	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) /\ B R C ) -> -. B = C )
8	7	3expia 1208	. . 3 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> ( B R C -> -. B = C ) )
9		so2nr 4368	. . . 4 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
10		imnan 692	. . . 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 754	. 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 710   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4044   Or wor 4342

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-po 4343  df-iso 4344

This theorem is referenced by:  sotritric  4371