Theorem po2nr 5572

Description: A partial order has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)

Assertion

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

Proof of Theorem po2nr

Step	Hyp	Ref	Expression
1		poirr 5570	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
2	1	adantrr 730	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵)
3		potr 5571	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))
4	3	3exp2 1373	. . . . 5 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
5	4	com34 92	. . . 4 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
6	5	pm2.43d 54	. . 3 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))
7	6	imp32 424	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))
8	2, 7	mtod 201	1 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2144   class class class wbr 5102   Po wpo 5556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-po 5558

This theorem is used by:  po3nr  5573  so2nr  5586  soisoi  7336  poxp2  8160  poxp3  8167  wemaplem2  9542  pospo  18518  poprelb  48682