Theorem sotri 5158

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 4802	. . . 4 |- ( A R B -> ( A e. S /\ B e. S ) )
3	2	simpld 112	. . 3 |- ( A R B -> A e. S )
4	1	brel 4802	. . 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 4439	. . . 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 2203   C_ wss 3211   class class class wbr 4109   Or wor 4416   X. cxp 4747

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-po 4417  df-iso 4418  df-xp 4755

This theorem is referenced by:  son2lpi  5159  ltsonq  7713  lt2addnq  7719  lt2mulnq  7720  ltbtwnnqq  7730  prarloclemarch2  7734  genplt2i  7825  addlocprlemgt  7849  nqprloc  7860  prmuloclemcalc  7880  ltsopr  7911  ltexprlemopl  7916  ltexprlemopu  7918  ltexprlemru  7927  prplnqu  7935  recexprlemlol  7941  recexprlemupu  7943  recexprlemdisj  7945  recexprlemss1l  7950  recexprlemss1u  7951  cauappcvgprlemopl  7961  cauappcvgprlemlol  7962  cauappcvgprlemupu  7964  cauappcvgprlemladdfu  7969  caucvgprlemk  7980  caucvgprlemnkj  7981  caucvgprlemnbj  7982  caucvgprlemm  7983  caucvgprlemopl  7984  caucvgprlemlol  7985  caucvgprlemupu  7987  caucvgprlemloc  7990  caucvgprlemladdfu  7992  caucvgprprlemk  7998  caucvgprprlemloccalc  7999  caucvgprprlemnkltj  8004  caucvgprprlemnkeqj  8005  caucvgprprlemnjltk  8006  caucvgprprlemnbj  8008  caucvgprprlemml  8009  caucvgprprlemopl  8012  caucvgprprlemlol  8013  caucvgprprlemupu  8015  lttrsr  8077  addgt0sr  8090  archsr  8097  caucvgsrlemcl  8104  caucvgsrlemfv  8106  suplocsrlemb  8121  suplocsrlempr  8122  suplocsrlem  8123  axpre-lttrn  8199