Theorem sotri 5130

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 4776	. . . 4 |- ( A R B -> ( A e. S /\ B e. S ) )
3	2	simpld 112	. . 3 |- ( A R B -> A e. S )
4	1	brel 4776	. . 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 4413	. . . 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 3198   class class class wbr 4086   Or wor 4390   X. cxp 4721

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 4205  ax-pow 4262  ax-pr 4297

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-po 4391  df-iso 4392  df-xp 4729

This theorem is referenced by:  son2lpi  5131  ltsonq  7608  lt2addnq  7614  lt2mulnq  7615  ltbtwnnqq  7625  prarloclemarch2  7629  genplt2i  7720  addlocprlemgt  7744  nqprloc  7755  prmuloclemcalc  7775  ltsopr  7806  ltexprlemopl  7811  ltexprlemopu  7813  ltexprlemru  7822  prplnqu  7830  recexprlemlol  7836  recexprlemupu  7838  recexprlemdisj  7840  recexprlemss1l  7845  recexprlemss1u  7846  cauappcvgprlemopl  7856  cauappcvgprlemlol  7857  cauappcvgprlemupu  7859  cauappcvgprlemladdfu  7864  caucvgprlemk  7875  caucvgprlemnkj  7876  caucvgprlemnbj  7877  caucvgprlemm  7878  caucvgprlemopl  7879  caucvgprlemlol  7880  caucvgprlemupu  7882  caucvgprlemloc  7885  caucvgprlemladdfu  7887  caucvgprprlemk  7893  caucvgprprlemloccalc  7894  caucvgprprlemnkltj  7899  caucvgprprlemnkeqj  7900  caucvgprprlemnjltk  7901  caucvgprprlemnbj  7903  caucvgprprlemml  7904  caucvgprprlemopl  7907  caucvgprprlemlol  7908  caucvgprprlemupu  7910  lttrsr  7972  addgt0sr  7985  archsr  7992  caucvgsrlemcl  7999  caucvgsrlemfv  8001  suplocsrlemb  8016  suplocsrlempr  8017  suplocsrlem  8018  axpre-lttrn  8094