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Definition df-nzr 19487
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.)
Assertion
Ref Expression
df-nzr NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}

Detailed syntax breakdown of Definition df-nzr
StepHypRef Expression
1 cnzr 19486 . 2 class NzRing
2 vr . . . . . 6 setvar 𝑟
32cv 1636 . . . . 5 class 𝑟
4 cur 18723 . . . . 5 class 1r
53, 4cfv 6111 . . . 4 class (1r𝑟)
6 c0g 16325 . . . . 5 class 0g
73, 6cfv 6111 . . . 4 class (0g𝑟)
85, 7wne 2989 . . 3 wff (1r𝑟) ≠ (0g𝑟)
9 crg 18769 . . 3 class Ring
108, 2, 9crab 3111 . 2 class {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
111, 10wceq 1637 1 wff NzRing = {𝑟 ∈ Ring ∣ (1r𝑟) ≠ (0g𝑟)}
Colors of variables: wff setvar class
This definition is referenced by:  isnzr  19488
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