Theorem sotri 3429

Description: A strict order relation is a transitive relation.

Hypotheses

Ref	Expression
soi.1	|- A e. V
soi.2	|- R Or S
soi.3	|- R (_ (S X. S)
sotri.4	|- B e. V
sotri.5	|- C e. V

Assertion

Ref	Expression
sotri	|- ((ARB /\ BRC) -> ARC)

Proof of Theorem sotri

Step	Hyp	Ref	Expression
1		id 59	. . . . . 6 |- ((A e. S /\ B e. S /\ C e. S) -> (A e. S /\ B e. S /\ C e. S))
2	1	3exp 830	. . . . 5 |- (A e. S -> (B e. S -> (C e. S -> (A e. S /\ B e. S /\ C e. S))))
3	2	a1dd 42	. . . 4 |- (A e. S -> (B e. S -> (B e. S -> (C e. S -> (A e. S /\ B e. S /\ C e. S)))))
4	3	imp43 370	. . 3 |- (((A e. S /\ B e. S) /\ (B e. S /\ C e. S)) -> (A e. S /\ B e. S /\ C e. S))
5		sotri.4	. . . 4 |- B e. V
6		soi.3	. . . 4 |- R (_ (S X. S)
7	5, 6	brel 3213	. . 3 |- (ARB -> (A e. S /\ B e. S))
8		sotri.5	. . . 4 |- C e. V
9	8, 6	brel 3213	. . 3 |- (BRC -> (B e. S /\ C e. S))
10	4, 7, 9	syl2an 454	. 2 |- ((ARB /\ BRC) -> (A e. S /\ B e. S /\ C e. S))
11		soi.2	. . 3 |- R Or S
12		sotr 2847	. . 3 |- ((R Or S /\ (A e. S /\ B e. S /\ C e. S)) -> ((ARB /\ BRC) -> ARC))
13	11, 12	mpan 693	. 2 |- ((A e. S /\ B e. S /\ C e. S) -> ((ARB /\ BRC) -> ARC))
14	10, 13	mpcom 49	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   e. wcel 955  Vcvv 1802   (_ wss 2037   class class class wbr 2609   Or wor 2830   X. cxp 3158

This theorem is referenced by:  son2lpi 3430  ltsopq 5047  ltrpq 5057  1pr 5089  prlem934 5111  ltexprlem4 5117  reclem2pr 5129  reclem4pr 5131  ltsosr 5175  addgt0sr 5185  suppsr2 5195  suppsr3 5196  ltsor 5233

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-po 2831  df-so 2841  df-xp 3174

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