Theorem po2nr 4408

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 4406	. . 3 |- ( ( R Po A /\ B e. A ) -> -. B R B )
2	1	adantrr 479	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3		potr 4407	. . . . . 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 1251	. . . . 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 257	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> ( ( B R C /\ C R B ) -> B R B ) )
8	2, 7	mtod 669	1 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2201   class class class wbr 4089   Po wpo 4393

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-v 2803  df-un 3203  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-po 4395

This theorem is referenced by:  po3nr  4409  so2nr  4420  tridc  7094