Theorem po2nr 5382

Description: A partial order relation 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 5380	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
2	1	adantrr 713	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵)
3		potr 5381	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))
4	3	3exp2 1347	. . . . 5 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
5	4	com34 91	. . . 4 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
6	5	pm2.43d 53	. . 3 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))
7	6	imp32 419	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))
8	2, 7	mtod 199	1 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2083   class class class wbr 4968   Po wpo 5367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-po 5369

This theorem is referenced by:  po3nr  5383  so2nr  5394  soisoi  6951  wemaplem2  8864  pospo  17416  poprelb  43190