Theorem sotri2 5038

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 999	. 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 4690	. . . . . 6 |- ( B R C -> ( B e. S /\ C e. S ) )
4	3	3ad2ant3 1021	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B e. S /\ C e. S ) )
5		simp1 998	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> A e. S )
6		df-3an 981	. . . . 5 |- ( ( B e. S /\ C e. S /\ A e. S ) <-> ( ( B e. S /\ C e. S ) /\ A e. S ) )
7	4, 5, 6	sylanbrc 417	. . . 4 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B e. S /\ C e. S /\ A e. S ) )
8		simp3 1000	. . . 4 |- ( ( A e. S /\ -. B R A /\ B R C ) -> B R C )
9		soi.1	. . . . 5 |- R Or S
10		sowlin 4332	. . . . 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 424	. . . 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 725	. 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 104   \/ wo 709   /\ w3a 979   e. wcel 2158   C_ wss 3141   class class class wbr 4015   Or wor 4307   X. cxp 4636

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-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-iso 4309  df-xp 4644

This theorem is referenced by: (None)