Theorem po2nr 4400

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 4398	. . 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 4399	. . . . . 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 1249	. . . . 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 667	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 2200   class class class wbr 4083   Po wpo 4385

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-po 4387

This theorem is referenced by:  po3nr  4401  so2nr  4412  tridc  7061