Theorem sotri3 4723

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 906	. 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 4392	. . . . 5 |- ( A R B -> ( A e. S /\ B e. S ) )
4	3	3ad2ant2 926	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S ) )
5		simp1 904	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> C e. S )
6		df-3an 887	. . . 4 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ C e. S ) )
7	4, 5, 6	sylanbrc 394	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S /\ C e. S ) )
8		simp2 905	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R B )
9		soi.1	. . . 4 |- R Or S
10		sowlin 4057	. . . 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 400	. . 3 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A R B -> ( A R C \/ C R B ) ) )
12	7, 8, 11	sylc 56	. 2 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A R C \/ C R B ) )
13	1, 12	ecased 1239	1 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ wo 629   /\ w3a 885   e. wcel 1393   C_ wss 2917   class class class wbr 3764   Or wor 4032   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-iso 4034  df-xp 4351

This theorem is referenced by: (None)