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Mirrors > Home > ILE Home > Th. List > isidom | GIF version |
Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.) |
Ref | Expression |
---|---|
isidom | ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-idom 13734 | . 2 ⊢ IDomn = (CRing ∩ Domn) | |
2 | 1 | elin2 3347 | 1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2164 CRingccrg 13471 Domncdomn 13730 IDomncidom 13731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-in 3159 df-idom 13734 |
This theorem is referenced by: znidom 14116 znidomb 14117 |
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