Theorem isdmn 33023

Description: The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)

Assertion

Ref	Expression
isdmn	⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Proof of Theorem isdmn

Step	Hyp	Ref	Expression
1		df-dmn 33018	. 2 ⊢ Dmn = (PrRing ∩ Com2)
2	1	elin2 3763	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Com2ccm2 32958  PrRingcprrng 33015  Dmncdmn 33016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-dmn 33018

This theorem is referenced by:  isdmn2  33024