Theorem so2nr 4357

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 4349	. 2 |- ( R Or A -> R Po A )
2		po2nr 4345	. 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 2167   class class class wbr 4034   Po wpo 4330   Or wor 4331

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-po 4332  df-iso 4333

This theorem is referenced by:  sotricim  4359  cauappcvgprlemdisj  7735  cauappcvgprlemladdru  7740  cauappcvgprlemladdrl  7741  caucvgprlemnbj  7751  caucvgprprlemnbj  7777  suplocexprlemmu  7802  ltnsym2  8134