Theorem so2nr 4368

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 4360	. 2 |- ( R Or A -> R Po A )
2		po2nr 4356	. 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 2176   class class class wbr 4044   Po wpo 4341   Or wor 4342

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-po 4343  df-iso 4344

This theorem is referenced by:  sotricim  4370  cauappcvgprlemdisj  7764  cauappcvgprlemladdru  7769  cauappcvgprlemladdrl  7770  caucvgprlemnbj  7780  caucvgprprlemnbj  7806  suplocexprlemmu  7831  ltnsym2  8163