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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isdmn | Structured version Visualization version GIF version |
Description: The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
isdmn | ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dmn 36511 | . 2 ⊢ Dmn = (PrRing ∩ Com2) | |
2 | 1 | elin2 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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