Theorem so2nr 4323

Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)

Assertion

Ref	Expression
so2nr	|- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Proof of Theorem so2nr

Step	Hyp	Ref	Expression
1		sopo 4315	. 2 |- ( R Or A -> R Po A )
2		po2nr 4311	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )
3	1, 2	sylan 283	1 |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2148   class class class wbr 4005   Po wpo 4296   Or wor 4297

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-po 4298  df-iso 4299

This theorem is referenced by:  sotricim  4325  cauappcvgprlemdisj  7652  cauappcvgprlemladdru  7657  cauappcvgprlemladdrl  7658  caucvgprlemnbj  7668  caucvgprprlemnbj  7694  suplocexprlemmu  7719  ltnsym2  8050