Theorem po2nr 5048

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 5046	. . 3 |- ( ( R Po A /\ B e. A ) -> -. B R B )
2	1	adantrr 753	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. B R B )
3		potr 5047	. . . . . 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 1285	. . . . 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 91	. . . 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 53	. . 3 |- ( R Po A -> ( B e. A -> ( C e. A -> ( ( B R C /\ C R B ) -> B R B ) ) ) )
7	6	imp32 449	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> ( ( B R C /\ C R B ) -> B R B ) )
8	2, 7	mtod 189	1 |- ( ( R Po A /\ ( B e. A /\ C e. A ) ) -> -. ( B R C /\ C R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   class class class wbr 4653   Po wpo 5033

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

This theorem is referenced by:  po3nr  5049  so2nr  5059  soisoi  6578  wemaplem2  8452  pospo  16973