Theorem poirr 2841

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 2840	. . . 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 776	. . 3 |- ((B e. A /\ B e. A /\ B e. A) <-> ((B e. A /\ B e. A) /\ B e. A))
5		anabs1 492	. . 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 774   e. wcel 957   class class class wbr 2615   Po wpo 2834

This theorem is referenced by:  po2nr 2843  sonr 2851  zorn2lem3 4773

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-v 1809  df-un 2047  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-po 2836

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