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Metamath Proof Explorer

Theorem lt0neg2d

Description: Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016)

		Ref	Expression
	Hypothesis	leidd.1	$⊢ φ \to A \in ℝ$
	Assertion	lt0neg2d	$⊢ φ \to (0 < A \leftrightarrow - A < 0)$

Step	Hyp	Ref	Expression
1		leidd.1	$⊢ φ \to A \in ℝ$
2		lt0neg2	$⊢ A \in ℝ \to (0 < A \leftrightarrow - A < 0)$
3	1 2	syl	$⊢ φ \to (0 < A \leftrightarrow - A < 0)$