Theorem sotri3 5100

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 1002	. 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 4745	. . . . 5 |- ( A R B -> ( A e. S /\ B e. S ) )
4	3	3ad2ant2 1022	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S ) )
5		simp1 1000	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> C e. S )
6		df-3an 983	. . . 4 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ C e. S ) )
7	4, 5, 6	sylanbrc 417	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S /\ C e. S ) )
8		simp2 1001	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R B )
9		soi.1	. . . 4 |- R Or S
10		sowlin 4385	. . . 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 424	. . 3 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( A R B -> ( A R C \/ C R B ) ) )
12	7, 8, 11	sylc 62	. 2 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A R C \/ C R B ) )
13	1, 12	ecased 1362	1 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   e. wcel 2178   C_ wss 3174   class class class wbr 4059   Or wor 4360   X. cxp 4691

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-iso 4362  df-xp 4699

This theorem is referenced by: (None)