Theorem so3nr 2854

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

Assertion

Ref	Expression
so3nr	|- ((R Or A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRC /\ CRD /\ DRB))

Proof of Theorem so3nr

Step	Hyp	Ref	Expression
1		po3nr 2843	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRC /\ CRD /\ DRB))
2		sopo 2846	. 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 /\ DRB))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 956   class class class wbr 2614   Po wpo 2833   Or wor 2834

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-po 2835  df-so 2845

Copyright terms: Public domain