Theorem sotri2 4944

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 4599	. . . . . 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 4250	. . . . 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 1481   C_ wss 3076   class class class wbr 3937   Or wor 4225   X. cxp 4545

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-iso 4227  df-xp 4553

This theorem is referenced by: (None)