Theorem sotr 2862

Description: A strict order relation is a transitive relation.

Assertion

Ref	Expression
sotr	|- ((R Or A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))

Proof of Theorem sotr

Step	Hyp	Ref	Expression
1		potr 2852	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))
2		sopo 2857	. 2 |- (R Or A -> R Po A)
3	1, 2	sylan 450	1 |- ((R Or A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   e. wcel 960   class class class wbr 2624   Po wpo 2844   Or wor 2845

This theorem is referenced by:  wetrep 2948  sotri 3449  pre-axlttrn 5300

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-po 2846  df-so 2856

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