Theorem sotri2 5141

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 1025	. 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 4784	. . . . . 6 |- ( B R C -> ( B e. S /\ C e. S ) )
4	3	3ad2ant3 1047	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> ( B e. S /\ C e. S ) )
5		simp1 1024	. . . . 5 |- ( ( A e. S /\ -. B R A /\ B R C ) -> A e. S )
6		df-3an 1007	. . . . 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 1026	. . . 4 |- ( ( A e. S /\ -. B R A /\ B R C ) -> B R C )
9		soi.1	. . . . 5 |- R Or S
10		sowlin 4423	. . . . 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 732	. 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 716   /\ w3a 1005   e. wcel 2202   C_ wss 3201   class class class wbr 4093   Or wor 4398   X. cxp 4729

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 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-iso 4400  df-xp 4737

This theorem is referenced by: (None)