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Metamath Proof Explorer

Theorem nnneneg

Description: No positive integer is equal to its negation. (Contributed by AV, 20-Jun-2023)

		Ref	Expression
	Assertion	nnneneg	$⊢ A \in ℕ \to A \neq - A$

Step	Hyp	Ref	Expression
1		nnne0	$⊢ A \in ℕ \to A \neq 0$
2	1	neneqd	$⊢ A \in ℕ \to \neg A = 0$
3		nncn	$⊢ A \in ℕ \to A \in ℂ$
4	3	eqnegd	$⊢ A \in ℕ \to (A = - A \leftrightarrow A = 0)$
5	2 4	mtbird	$⊢ A \in ℕ \to \neg A = - A$
6	5	neqned	$⊢ A \in ℕ \to A \neq - A$