Theorem sotri2 4982

Description: A transitivity relation. (Read -. B < A and B < C 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
sotri2	|- ( ( A e. S /\ -. B R A /\ B R C ) -> A R C )

Proof of Theorem sotri2

Step	Hyp	Ref	Expression
1		simp2 983	. 2 |- ( ( A e. S /\ -. B R A /\ B R C ) -> -. B R A )
2		soi.2	. . . . . . 7 |- R C_ ( S X. S )
3	2	brel 4637	. . . . . 6 |- ( B R C -> ( B e. S /\ C e. S ) )
4	3	3ad2ant3 1005	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B e. S /\ C e. S ) )
5		simp1 982	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> A e. S )
6		df-3an 965	. . . . 5 |- ( ( B e. S /\ C e. S /\ A e. S ) <-> ( ( B e. S /\ C e. S ) /\ A e. S ) )
7	4, 5, 6	sylanbrc 414	. . . 4 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B e. S /\ C e. S /\ A e. S ) )
8		simp3 984	. . . 4 |- ( ( A e. S /\ -. B R A /\ B R C ) -> B R C )
9		soi.1	. . . . 5 |- R Or S
10		sowlin 4280	. . . . 5 |- ( ( R Or S /\ ( B e. S /\ C e. S /\ A e. S ) ) -> ( B R C -> ( B R A \/ A R C ) ) )
11	9, 10	mpan 421	. . . 4 |- ( ( B e. S /\ C e. S /\ A e. S ) -> ( B R C -> ( B R A \/ A R C ) ) )
12	7, 8, 11	sylc 62	. . 3 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B R A \/ A R C ) )
13	12	ord 714	. 2 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( -. B R A -> A R C ) )
14	1, 13	mpd 13	1 |- ( ( A e. S /\ -. B R A /\ B R C ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 963   e. wcel 2128   C_ wss 3102   class class class wbr 3965   Or wor 4255   X. cxp 4583

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-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-iso 4257  df-xp 4591

This theorem is referenced by: (None)