Theorem sotri3 5181

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 1030	. 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 4822	. . . . 5 |- ( A R B -> ( A e. S /\ B e. S ) )
4	3	3ad2ant2 1050	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S ) )
5		simp1 1028	. . . 4 |- ( ( C e. S /\ A R B /\ -. C R B ) -> C e. S )
6		df-3an 1011	. . . 4 |- ( ( A e. S /\ B e. S /\ C e. S ) <-> ( ( A e. S /\ B e. S ) /\ C e. S ) )
7	4, 5, 6	sylanbrc 421	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> ( A e. S /\ B e. S /\ C e. S ) )
8		simp2 1029	. . 3 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R B )
9		soi.1	. . . 4 |- R Or S
10		sowlin 4460	. . . 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 428	. . 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 1390	1 |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   e. wcel 2209   C_ wss 3220   class class class wbr 4125   Or wor 4435   X. cxp 4767

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 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-iso 4437  df-xp 4775

This theorem is referenced by: (None)