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Mirrors > Home > ILE Home > Th. List > df-nzr | GIF version |
Description: A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
df-nzr | ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnzr 13254 | . 2 class NzRing | |
2 | vr | . . . . . 6 setvar 𝑟 | |
3 | 2 | cv 1352 | . . . . 5 class 𝑟 |
4 | cur 13073 | . . . . 5 class 1r | |
5 | 3, 4 | cfv 5215 | . . . 4 class (1r‘𝑟) |
6 | c0g 12693 | . . . . 5 class 0g | |
7 | 3, 6 | cfv 5215 | . . . 4 class (0g‘𝑟) |
8 | 5, 7 | wne 2347 | . . 3 wff (1r‘𝑟) ≠ (0g‘𝑟) |
9 | crg 13110 | . . 3 class Ring | |
10 | 8, 2, 9 | crab 2459 | . 2 class {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} |
11 | 1, 10 | wceq 1353 | 1 wff NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} |
Colors of variables: wff set class |
This definition is referenced by: isnzr 13256 nzrring 13258 |
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