Theorem sotri3 4743

Description: A transitivity relation. (Read A < B and -. C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	|- R Or S
soi.2	|- R C_ ( S X. S )

Assertion

Ref	Expression
sotri3	|- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Proof of Theorem sotri3

Step	Hyp	Ref	Expression
1		simp3 940	. 2 |- ( ( C e. S /\ A R B /\ -. C R B ) -> -. C R B )
2		soi.2	. . . . . 6 |- R C_ ( S X. S )
3	2	brel 4410	. . . . 5 |- ( A R B -> ( A e. S /\ B e. S ) )
4	3	3ad2ant2 960	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S ) )
5		simp1 938	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> C e. S )
6		df-3an 921	. . . 4 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ C e. S ) )
7	4, 5, 6	sylanbrc 408	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S /\ C e. S ) )
8		simp2 939	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R B )
9		soi.1	. . . 4 |- R Or S
10		sowlin 4075	. . . 4 |- ( ( R Or S /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( A R B -> ( A R C \/ C R B ) ) )
11	9, 10	mpan 414	. . 3 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A R B -> ( A R C \/ C R B ) ) )
12	7, 8, 11	sylc 61	. 2 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A R C \/ C R B ) )
13	1, 12	ecased 1280	1 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661   /\ w3a 919   e. wcel 1433   C_ wss 2973   class class class wbr 3785   Or wor 4050   X. cxp 4361

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-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-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

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-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-iso 4052  df-xp 4369

This theorem is referenced by: (None)