Theorem so3nr 5060

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 5052	. 2 |- ( R Or A -> R Po A )
2		po3nr 5049	. 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 488	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 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653   Po wpo 5033   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-po 5035  df-so 5036

This theorem is referenced by: (None)