Theorem po2nr 2135

Description: A partial order relation has no 2-cycle loops.

Assertion

Ref	Expression
po2nr	⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A)) → ¬ (BRC ∧ CRB))

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 2133	. . 3 ⊢ ((R Po A ∧ B ∈ A) → ¬ BRB)
2	1	adantrr 312	. 2 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A)) → ¬ BRB)
3		potr 2134	. . . . . . 7 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A ∧ B ∈ A)) → ((BRC ∧ CRB) → BRB))
4		df-3an 583	. . . . . . 7 ⊢ ((B ∈ A ∧ C ∈ A ∧ B ∈ A) ↔ ((B ∈ A ∧ C ∈ A) ∧ B ∈ A))
5	3, 4	sylan2br 348	. . . . . 6 ⊢ ((R Po A ∧ ((B ∈ A ∧ C ∈ A) ∧ B ∈ A)) → ((BRC ∧ CRB) → BRB))
6	5	exp44 302	. . . . 5 ⊢ (R Po A → (B ∈ A → (C ∈ A → (B ∈ A → ((BRC ∧ CRB) → BRB)))))
7	6	com34 36	. . . 4 ⊢ (R Po A → (B ∈ A → (B ∈ A → (C ∈ A → ((BRC ∧ CRB) → BRB)))))
8	7	pm2.43d 59	. . 3 ⊢ (R Po A → (B ∈ A → (C ∈ A → ((BRC ∧ CRB) → BRB))))
9	8	imp32 281	. 2 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A)) → ((BRC ∧ CRB) → BRB))
10	2, 9	mtod 95	1 ⊢ ((R Po A ∧ (B ∈ A ∧ C ∈ A)) → ¬ (BRC ∧ CRB))

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:  po3nr 2136  so2nr 2146

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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