Theorem poirr 2836

Description: A partial order relation is irreflexive.

Assertion

Ref	Expression
poirr	|- ((R Po A /\ B e. A) -> -. BRB)

Proof of Theorem poirr

Step	Hyp	Ref	Expression
1		pocl 2835	. . . 4 |- (R Po A -> ((B e. A /\ B e. A /\ B e. A) -> (-. BRB /\ ((BRB /\ BRB) -> BRB))))
2	1	imp 350	. . 3 |- ((R Po A /\ (B e. A /\ B e. A /\ B e. A)) -> (-. BRB /\ ((BRB /\ BRB) -> BRB)))
3	2	pm3.26d 321	. 2 |- ((R Po A /\ (B e. A /\ B e. A /\ B e. A)) -> -. BRB)
4		df-3an 775	. . 3 |- ((B e. A /\ B e. A /\ B e. A) <-> ((B e. A /\ B e. A) /\ B e. A))
5		anabs1 491	. . 3 |- (((B e. A /\ B e. A) /\ B e. A) <-> (B e. A /\ B e. A))
6		anidm 432	. . 3 |- ((B e. A /\ B e. A) <-> B e. A)
7	4, 5, 6	3bitrr 178	. 2 |- (B e. A <-> (B e. A /\ B e. A /\ B e. A))
8	3, 7	sylan2b 452	1 |- ((R Po A /\ B e. A) -> -. BRB)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 773   e. wcel 955   class class class wbr 2609   Po wpo 2829

This theorem is referenced by:  po2nr 2838  sonr 2846  zorn2lem3 4762

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-po 2831

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