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Metamath Proof Explorer

Theorem ltnsym2

Description: 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005) (Proof shortened by Andrew Salmon, 19-Nov-2011)

		Ref	Expression
	Assertion	ltnsym2	$⊢ (A \in ℝ \land B \in ℝ) \to \neg (A < B \land B < A)$

Step	Hyp	Ref	Expression
1		ltso	$⊢ < Or ℝ$
2		so2nr	$⊢ (< Or ℝ \land (A \in ℝ \land B \in ℝ)) \to \neg (A < B \land B < A)$
3	1 2	mpan	$⊢ (A \in ℝ \land B \in ℝ) \to \neg (A < B \land B < A)$