Theorem isdmn 38737

Description: Obsolete theorem, use isidom2 49141 instead. The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem isdmn

Step	Hyp	Ref	Expression
1		df-dmn 38732	. 2 ⊢ Dmn = (PrRing ∩ Com2)
2	1	elin2 4159	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  Com2ccm2 38672  PrRingcprrng 38729  Dmncdmn 38730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-in 3915  df-dmn 38732

This theorem is used by:  isdmn2  38738