Theorem sotrieq2 5534

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 5533	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
2		ioran 981	. 2 ⊢ (¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵) ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))
3	1, 2	bitrdi 287	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1542   ∈ wcel 2110   class class class wbr 5079   Or wor 5503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-po 5504  df-so 5505

This theorem is referenced by:  fisupg  9040  supmo  9189  infmo  9232  fiinfg  9236  lttri3  11059  xrlttri3  12876  nosupbnd1lem2  33908  noinfbnd1lem2  33923  slttrieq2  33949  wessf1ornlem  42692