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Theorem isdmn 36516
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 36511 . 2 Dmn = (PrRing ∩ Com2)
21elin2 4158 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  Com2ccm2 36451  PrRingcprrng 36508  Dmncdmn 36509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-in 3918  df-dmn 36511
This theorem is referenced by:  isdmn2  36517
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