Theorem po2nr 2847

Description: A partial order relation has no 2-cycle loops.

Assertion

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

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 2845	. . 3 |- ((R Po A /\ B e. A) -> -. BRB)
2	1	adantrr 395	. 2 |- ((R Po A /\ (B e. A /\ C e. A)) -> -. BRB)
3		potr 2846	. . . . . 6 |- ((R Po A /\ (B e. A /\ C e. A /\ B e. A)) -> ((BRC /\ CRB) -> BRB))
4	3	3exp2 851	. . . . 5 |- (R Po A -> (B e. A -> (C e. A -> (B e. A -> ((BRC /\ CRB) -> BRB)))))
5	4	com34 36	. . . 4 |- (R Po A -> (B e. A -> (B e. A -> (C e. A -> ((BRC /\ CRB) -> BRB)))))
6	5	pm2.43d 65	. . 3 |- (R Po A -> (B e. A -> (C e. A -> ((BRC /\ CRB) -> BRB))))
7	6	imp32 363	. 2 |- ((R Po A /\ (B e. A /\ C e. A)) -> ((BRC /\ CRB) -> BRB))
8	2, 7	mtod 108	1 |- ((R Po A /\ (B e. A /\ C e. A)) -> -. (BRC /\ CRB))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 958   class class class wbr 2619   Po wpo 2838

This theorem is referenced by:  po3nr 2848  so2nr 2858

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-po 2840

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