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Metamath Proof Explorer

Theorem lttri3

Description: Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999)

		Ref	Expression
	Assertion	lttri3	$⊢ (A \in ℝ \land B \in ℝ) \to (A = B \leftrightarrow (\neg A < B \land \neg B < A))$

Step	Hyp	Ref	Expression
1		ltso	$⊢ < Or ℝ$
2		sotrieq2	$⊢ (< Or ℝ \land (A \in ℝ \land B \in ℝ)) \to (A = B \leftrightarrow (\neg A < B \land \neg B < A))$
3	1 2	mpan	$⊢ (A \in ℝ \land B \in ℝ) \to (A = B \leftrightarrow (\neg A < B \land \neg B < A))$