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Metamath Proof Explorer

Theorem xrlttri3

Description: Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006)

		Ref	Expression
	Assertion	xrlttri3	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (A = B \leftrightarrow (\neg A < B \land \neg B < A))$

Step	Hyp	Ref	Expression
1		xrltso	$⊢ < Or ℝ^{*}$
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	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (A = B \leftrightarrow (\neg A < B \land \neg B < A))$