Theorem sotrieq2 2868

Description: Trichotomy law for strict order relation.

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 2867	. 2 ⊢ ((R Or A ⋀ (B ∈ A ⋀ C ∈ A)) → (B = C ↔ ¬ (BRC ⋁ CRB)))
2		ioran 306	. 2 ⊢ (¬ (BRC ⋁ CRB) ↔ (¬ BRC ⋀ ¬ CRB))
3	1, 2	syl6bb 538	1 ⊢ ((R Or A ⋀ (B ∈ A ⋀ C ∈ A)) → (B = C ↔ (¬ BRC ⋀ ¬ CRB)))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   = wceq 958   ∈ wcel 960   class class class wbr 2624   Or wor 2845

This theorem is referenced by:  supmo 4585  supmax 4598  lttri3t 5526  xrlttri3t 5568

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-po 2846  df-so 2856

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