Theorem so3nr 4368

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 4359	. 2 |- ( R Or A -> R Po A )
2		po3nr 4356	. 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 980   e. wcel 2175   class class class wbr 4043   Po wpo 4340   Or wor 4341

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-po 4342  df-iso 4343

This theorem is referenced by: (None)