Theorem sotri 5139

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 4784	. . . 4 |- ( A R B -> ( A e. S /\ B e. S ) )
3	2	simpld 112	. . 3 |- ( A R B -> A e. S )
4	1	brel 4784	. . 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 4421	. . . 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 1231	. 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 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-in1 619  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-po 4399  df-iso 4400  df-xp 4737

This theorem is referenced by:  son2lpi  5140  ltsonq  7678  lt2addnq  7684  lt2mulnq  7685  ltbtwnnqq  7695  prarloclemarch2  7699  genplt2i  7790  addlocprlemgt  7814  nqprloc  7825  prmuloclemcalc  7845  ltsopr  7876  ltexprlemopl  7881  ltexprlemopu  7883  ltexprlemru  7892  prplnqu  7900  recexprlemlol  7906  recexprlemupu  7908  recexprlemdisj  7910  recexprlemss1l  7915  recexprlemss1u  7916  cauappcvgprlemopl  7926  cauappcvgprlemlol  7927  cauappcvgprlemupu  7929  cauappcvgprlemladdfu  7934  caucvgprlemk  7945  caucvgprlemnkj  7946  caucvgprlemnbj  7947  caucvgprlemm  7948  caucvgprlemopl  7949  caucvgprlemlol  7950  caucvgprlemupu  7952  caucvgprlemloc  7955  caucvgprlemladdfu  7957  caucvgprprlemk  7963  caucvgprprlemloccalc  7964  caucvgprprlemnkltj  7969  caucvgprprlemnkeqj  7970  caucvgprprlemnjltk  7971  caucvgprprlemnbj  7973  caucvgprprlemml  7974  caucvgprprlemopl  7977  caucvgprprlemlol  7978  caucvgprprlemupu  7980  lttrsr  8042  addgt0sr  8055  archsr  8062  caucvgsrlemcl  8069  caucvgsrlemfv  8071  suplocsrlemb  8086  suplocsrlempr  8087  suplocsrlem  8088  axpre-lttrn  8164