Theorem isdmn 35331

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 35326	. 2 ⊢ Dmn = (PrRing ∩ Com2)
2	1	elin2 4173	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  Com2ccm2 35266  PrRingcprrng 35323  Dmncdmn 35324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-dmn 35326

This theorem is referenced by:  isdmn2  35332