Theorem sotrieq2 4987

Description: Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)

Assertion

Ref	Expression
sotrieq2	⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))

Proof of Theorem sotrieq2

Step	Hyp	Ref	Expression
1		sotrieq 4986	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
2		ioran 510	. 2 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
3	1, 2	syl6bb 275	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583   Or wor 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-po 4959  df-so 4960

This theorem is referenced by:  fisupg  8093  supmo  8241  infmo  8284  fiinfg  8288  lttri3  10000  xrlttri3  11852  slttrieq2  31073  wessf1ornlem  38366