Theorem so3nr 4387

Description: A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)

Assertion

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

Proof of Theorem so3nr

Step	Hyp	Ref	Expression
1		sopo 4378	. 2 |- ( R Or A -> R Po A )
2		po3nr 4375	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R C /\ C R D /\ D R B ) )
3	1, 2	sylan 283	1 |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R C /\ C R D /\ D R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 981   e. wcel 2178   class class class wbr 4059   Po wpo 4359   Or wor 4360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-po 4361  df-iso 4362

This theorem is referenced by: (None)