Theorem poirr 2133

Description: A partial order relation is irreflexive.

Assertion

Ref	Expression
poirr	⊢ ((R Po A ∧ B ∈ A) → ¬ BRB)

Proof of Theorem poirr

Step	Hyp	Ref	Expression
1		pocl 2132	. . . 4 ⊢ (R Po A → ((B ∈ A ∧ B ∈ A ∧ B ∈ A) → (¬ BRB ∧ ((BRB ∧ BRB) → BRB))))
2	1	imp 277	. . 3 ⊢ ((R Po A ∧ (B ∈ A ∧ B ∈ A ∧ B ∈ A)) → (¬ BRB ∧ ((BRB ∧ BRB) → BRB)))
3	2	pm3.26d 258	. 2 ⊢ ((R Po A ∧ (B ∈ A ∧ B ∈ A ∧ B ∈ A)) → ¬ BRB)
4		anidm 331	. . 3 ⊢ ((B ∈ A ∧ B ∈ A) ↔ B ∈ A)
5		df-3an 583	. . . 4 ⊢ ((B ∈ A ∧ B ∈ A ∧ B ∈ A) ↔ ((B ∈ A ∧ B ∈ A) ∧ B ∈ A))
6		anabs1 374	. . . 4 ⊢ (((B ∈ A ∧ B ∈ A) ∧ B ∈ A) ↔ (B ∈ A ∧ B ∈ A))
7	5, 6	bitr2 152	. . 3 ⊢ ((B ∈ A ∧ B ∈ A) ↔ (B ∈ A ∧ B ∈ A ∧ B ∈ A))
8	4, 7	bitr3 153	. 2 ⊢ (B ∈ A ↔ (B ∈ A ∧ B ∈ A ∧ B ∈ A))
9	3, 8	sylan2b 347	1 ⊢ ((R Po A ∧ B ∈ A) → ¬ BRB)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581   ∈ 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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