Theorem po3nr 2848

Description: A partial order relation has no 3-cycle loops.

Assertion

Ref	Expression
po3nr	|- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRC /\ CRD /\ DRB))

Proof of Theorem po3nr

Step	Hyp	Ref	Expression
1		po2nr 2847	. . 3 |- ((R Po A /\ (B e. A /\ D e. A)) -> -. (BRD /\ DRB))
2	1	3adantr2 807	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRD /\ DRB))
3		potr 2846	. . . 4 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))
4	3	anim1d 560	. . 3 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> (((BRC /\ CRD) /\ DRB) -> (BRD /\ DRB)))
5		df-3an 777	. . 3 |- ((BRC /\ CRD /\ DRB) <-> ((BRC /\ CRD) /\ DRB))
6	4, 5	syl5ib 206	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD /\ DRB) -> (BRD /\ DRB)))
7	2, 6	mtod 108	1 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRC /\ CRD /\ DRB))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775   e. wcel 958   class class class wbr 2619   Po wpo 2838

This theorem is referenced by:  so3nr 2859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-po 2840

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