Theorem po3nr 2136

Description: A partial order relation has no 3-cycle loops.

Assertion

Ref	Expression
po3nr	⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ D ∈ A)) → ¬ (BRC ∧ CRD ∧ DRB))

Proof of Theorem po3nr

Step	Hyp	Ref	Expression
1		po2nr 2135	. . 3 ⊢ ((R Po A ∧ (B ∈ A ∧ D ∈ A)) → ¬ (BRD ∧ DRB))
2		3simpb 592	. . 3 ⊢ ((B ∈ A ∧ C ∈ A ∧ D ∈ A) → (B ∈ A ∧ D ∈ A))
3	1, 2	sylan2 346	. 2 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ D ∈ A)) → ¬ (BRD ∧ DRB))
4		potr 2134	. . . 4 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ D ∈ A)) → ((BRC ∧ CRD) → BRD))
5	4	anim1d 432	. . 3 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ D ∈ A)) → (((BRC ∧ CRD) ∧ DRB) → (BRD ∧ DRB)))
6		df-3an 583	. . 3 ⊢ ((BRC ∧ CRD ∧ DRB) ↔ ((BRC ∧ CRD) ∧ DRB))
7	5, 6	syl5ib 181	. 2 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ D ∈ A)) → ((BRC ∧ CRD ∧ DRB) → (BRD ∧ DRB)))
8	3, 7	mtod 95	1 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ D ∈ A)) → ¬ (BRC ∧ CRD ∧ DRB))

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:  so3nr 2147

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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