Theorem sotri3 5009

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 994	. 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 4663	. . . . 5 |- ( A R B -> ( A e. S /\ B e. S ) )
4	3	3ad2ant2 1014	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S ) )
5		simp1 992	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> C e. S )
6		df-3an 975	. . . 4 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ C e. S ) )
7	4, 5, 6	sylanbrc 415	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S /\ C e. S ) )
8		simp2 993	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R B )
9		soi.1	. . . 4 |- R Or S
10		sowlin 4305	. . . 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 422	. . 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 1344	1 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 703   /\ w3a 973   e. wcel 2141   C_ wss 3121   class class class wbr 3989   Or wor 4280   X. cxp 4609

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-iso 4282  df-xp 4617

This theorem is referenced by: (None)