Theorem sotri 5124

Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	|- R Or S
soi.2	|- R C_ ( S X. S )

Assertion

Ref	Expression
sotri	|- ( ( A R B /\ B R C ) -> A R C )

Proof of Theorem sotri

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 |- R C_ ( S X. S )
2	1	brel 4771	. . . 4 |- ( A R B -> ( A e. S /\ B e. S ) )
3	2	simpld 112	. . 3 |- ( A R B -> A e. S )
4	1	brel 4771	. . 3 |- ( B R C -> ( B e. S /\ C e. S ) )
5	3, 4	anim12i 338	. 2 |- ( ( A R B /\ B R C ) -> ( A e. S /\ ( B e. S /\ C e. S ) ) )
6		soi.1	. . . 4 |- R Or S
7		sotr 4409	. . . 4 |- ( ( R Or S /\ ( A e. S /\ B e. S /\ C e. S ) ) -> ( ( A R B /\ B R C ) -> A R C ) )
8	6, 7	mpan 424	. . 3 |- ( ( A e. S /\ B e. S /\ C e. S ) -> ( ( A R B /\ B R C ) -> A R C ) )
9	8	3expb 1228	. 2 |- ( ( A e. S /\ ( B e. S /\ C e. S ) ) -> ( ( A R B /\ B R C ) -> A R C ) )
10	5, 9	mpcom 36	1 |- ( ( A R B /\ B R C ) -> A R C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   C_ wss 3197   class class class wbr 4083   Or wor 4386   X. cxp 4717

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-po 4387  df-iso 4388  df-xp 4725

This theorem is referenced by:  son2lpi  5125  ltsonq  7585  lt2addnq  7591  lt2mulnq  7592  ltbtwnnqq  7602  prarloclemarch2  7606  genplt2i  7697  addlocprlemgt  7721  nqprloc  7732  prmuloclemcalc  7752  ltsopr  7783  ltexprlemopl  7788  ltexprlemopu  7790  ltexprlemru  7799  prplnqu  7807  recexprlemlol  7813  recexprlemupu  7815  recexprlemdisj  7817  recexprlemss1l  7822  recexprlemss1u  7823  cauappcvgprlemopl  7833  cauappcvgprlemlol  7834  cauappcvgprlemupu  7836  cauappcvgprlemladdfu  7841  caucvgprlemk  7852  caucvgprlemnkj  7853  caucvgprlemnbj  7854  caucvgprlemm  7855  caucvgprlemopl  7856  caucvgprlemlol  7857  caucvgprlemupu  7859  caucvgprlemloc  7862  caucvgprlemladdfu  7864  caucvgprprlemk  7870  caucvgprprlemloccalc  7871  caucvgprprlemnkltj  7876  caucvgprprlemnkeqj  7877  caucvgprprlemnjltk  7878  caucvgprprlemnbj  7880  caucvgprprlemml  7881  caucvgprprlemopl  7884  caucvgprprlemlol  7885  caucvgprprlemupu  7887  lttrsr  7949  addgt0sr  7962  archsr  7969  caucvgsrlemcl  7976  caucvgsrlemfv  7978  suplocsrlemb  7993  suplocsrlempr  7994  suplocsrlem  7995  axpre-lttrn  8071