Theorem po2nr 4818

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

Assertion

Ref	Expression
po2nr

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 4816	. . 3
2	1	adantrr 716	. 2
3		potr 4817	. . . . . 6
4	3	3exp2 1214	. . . . 5
5	4	com34 83	. . . 4
6	5	pm2.43d 48	. . 3
7	6	imp32 433	. 2
8	2, 7	mtod 177	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  e.wcel 1818   class class class wbr 4452  Powpo 4803

This theorem is referenced by:  po3nr  4819  so2nr  4829  soisoi  6224  wemaplem2  7993  pospo  15603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-po 4805