Theorem sotri2 5001

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 988	. 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 4656	. . . . . 6 |- ( B R C -> ( B e. S /\ C e. S ) )
4	3	3ad2ant3 1010	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B e. S /\ C e. S ) )
5		simp1 987	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> A e. S )
6		df-3an 970	. . . . 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 989	. . . 4 |- ( ( A e. S /\ -. B R A /\ B R C ) -> B R C )
9		soi.1	. . . . 5 |- R Or S
10		sowlin 4298	. . . . 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 968   e. wcel 2136   C_ wss 3116   class class class wbr 3982   Or wor 4273   X. cxp 4602

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-iso 4275  df-xp 4610

This theorem is referenced by: (None)