Theorem isdmn 38093

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 38088	. 2 ⊢ Dmn = (PrRing ∩ Com2)
2	1	elin2 4153	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  Com2ccm2 38028  PrRingcprrng 38085  Dmncdmn 38086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3909  df-dmn 38088

This theorem is referenced by:  isdmn2  38094