Theorem isfld

Description: A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015)

	Ref	Expression
Assertion	isfld	$$⊢R\in \mathrm{Field}\leftrightarrow \left(R\in \mathrm{DivRing}\wedge R\in \mathrm{CRing}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-field	$$⊢\mathrm{Field}=\mathrm{DivRing}\cap \mathrm{CRing}$$
2	1	elin2	$$⊢R\in \mathrm{Field}\leftrightarrow \left(R\in \mathrm{DivRing}\wedge R\in \mathrm{CRing}\right)$$