Theorem poirr 2133

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 2132	. . . 4 |- (R Po A -> ((B e. A /\ B e. A /\ B e. A) -> (-. BRB /\ ((BRB /\ BRB) -> BRB))))
2	1	imp 277	. . 3 |- ((R Po A /\ (B e. A /\ B e. A /\ B e. A)) -> (-. BRB /\ ((BRB /\ BRB) -> BRB)))
3	2	pm3.26d 258	. 2 |- ((R Po A /\ (B e. A /\ B e. A /\ B e. A)) -> -. BRB)
4		anidm 331	. . 3 |- ((B e. A /\ B e. A) <-> B e. A)
5		df-3an 583	. . . 4 |- ((B e. A /\ B e. A /\ B e. A) <-> ((B e. A /\ B e. A) /\ B e. A))
6		anabs1 374	. . . 4 |- (((B e. A /\ B e. A) /\ B e. A) <-> (B e. A /\ B e. A))
7	5, 6	bitr2 152	. . 3 |- ((B e. A /\ B e. A) <-> (B e. A /\ B e. A /\ B e. A))
8	4, 7	bitr3 153	. 2 |- (B e. A <-> (B e. A /\ B e. A /\ B e. A))
9	3, 8	sylan2b 347	1 |- ((R Po A /\ B e. A) -> -. BRB)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   /\ w3a 581   e. wcel 1092   class class class wbr 2054   Po wpo 2058

This theorem is referenced by:  po2nr 2135  sonr 2143  zornlem3 3605

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128

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