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Theorem isdmn 36502
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
StepHypRef Expression
1 df-dmn 36497 . 2 Dmn = (PrRing ∩ Com2)
21elin2 4156 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wcel 2106  Com2ccm2 36437  PrRingcprrng 36494  Dmncdmn 36495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3446  df-in 3916  df-dmn 36497
This theorem is referenced by:  isdmn2  36503
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