Theorem so2nr 5576

Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)

Assertion

Ref	Expression
so2nr	⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Proof of Theorem so2nr

Step	Hyp	Ref	Expression
1		sopo 5567	. 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
2		po2nr 5562	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
3	1, 2	sylan 588	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∈ wcel 2136   class class class wbr 5094   Po wpo 5546   Or wor 5547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-po 5548  df-so 5549

This theorem is referenced by:  sotric  5578  somincom  6111  fisupg  9221  suppr  9408  infpr  9441  genpnnp  10953  ltnsym2  11272