Theorem crngprmringdom

Description: A commutative ring is a prime ring if and only if it is a domain. (Contributed by AV, 27-Jun-2026)

	Ref	Expression
Assertion	crngprmringdom	Could not format assertion : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> R e. Domn ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		crngprmringidom	Could not format ( R e. CRing -> ( R e. PrmRing <-> R e. IDomn ) ) : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> R e. IDomn ) ) with typecode |-
2		isidom	$$⊢R\in \mathrm{IDomn}\leftrightarrow \left(R\in \mathrm{CRing}\wedge R\in \mathrm{Domn}\right)$$
3	2	baib	$$⊢R\in \mathrm{CRing}\to \left(R\in \mathrm{IDomn}\leftrightarrow R\in \mathrm{Domn}\right)$$
4	1 3	bitrd	Could not format ( R e. CRing -> ( R e. PrmRing <-> R e. Domn ) ) : No typesetting found for |- ( R e. CRing -> ( R e. PrmRing <-> R e. Domn ) ) with typecode |-