Theorem sotr 2852

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 2842	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))
2		sopo 2847	. 2 |- (R Or A -> R Po A)
3	1, 2	sylan 448	1 |- ((R Or A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 957   class class class wbr 2615   Po wpo 2834   Or wor 2835

This theorem is referenced by:  wetrep 2938  sotri 3439  pre-axlttrn 5271

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-v 1809  df-un 2047  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-po 2836  df-so 2846

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