Theorem po2nr 4287

Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
po2nr	|- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 4285	. . 3 |- ( ( R Po A /\ B e. A ) -> -. B R B )
2	1	adantrr 471	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3		potr 4286	. . . . . 6 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ B e. A ) ) -> ( ( B R C /\ C R B ) -> B R B ) )
4	3	3exp2 1215	. . . . 5 |- ( R Po A -> ( B e. A -> ( C e. A -> ( B e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) ) )
5	4	com34 83	. . . 4 |- ( R Po A -> ( B e. A -> ( B e. A -> ( C e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) ) )
6	5	pm2.43d 50	. . 3 |- ( R Po A -> ( B e. A -> ( C e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) )
7	6	imp32 255	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> ( ( B R C /\ C R B ) -> B R B ) )
8	2, 7	mtod 653	1 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2136   class class class wbr 3982   Po wpo 4272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-po 4274

This theorem is referenced by:  po3nr  4288  so2nr  4299  tridc  6865