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Metamath Proof Explorer

Theorem nzrring

Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015) (Proof shortened by SN, 23-Feb-2025)

		Ref	Expression
	Assertion	nzrring	$⊢ R \in NzRing \to R \in Ring$

Step	Hyp	Ref	Expression
1		df-nzr	$⊢ NzRing = \{r \in Ring \| 1_{r} \neq 0_{r}\}$
2	1	ssrab3	$⊢ NzRing \subseteq Ring$
3	2	sseli	$⊢ R \in NzRing \to R \in Ring$