Theorem so2nr 5211

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 5204	. 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
2		po2nr 5200	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
3	1, 2	sylan 489	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 2139   class class class wbr 4804   Po wpo 5185   Or wor 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-po 5187  df-so 5188

This theorem is referenced by:  sotric  5213  somincom  5688  fisupg  8373  suppr  8542  fiinfg  8570  infpr  8574  genpnnp  10019  ltnsym2  10328