Theorem potr 2852

Description: A partial order relation is a transitive relation.

Assertion

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

Proof of Theorem potr

Step	Hyp	Ref	Expression
1		pocl 2850	. . 3 |- (R Po A -> ((B e. A /\ C e. A /\ D e. A) -> (-. BRB /\ ((BRC /\ CRD) -> BRD))))
2	1	imp 350	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> (-. BRB /\ ((BRC /\ CRD) -> BRD)))
3	2	pm3.27d 325	1 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 777   e. wcel 960   class class class wbr 2624   Po wpo 2844

This theorem is referenced by:  po2nr 2853  po3nr 2854  sotr 2862  zorn2lem7 4804

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

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