Definition df-nzr 13736

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 ∣ (1r‘𝑟) ≠ (0g‘𝑟)}

Detailed syntax breakdown of Definition df-nzr

Step	Hyp	Ref	Expression
1		cnzr 13735	. 2 class NzRing
2		vr	. . . . . 6 setvar 𝑟
3	2	cv 1363	. . . . 5 class 𝑟
4		cur 13515	. . . . 5 class 1r
5	3, 4	cfv 5258	. . . 4 class (1r‘𝑟)
6		c0g 12927	. . . . 5 class 0g
7	3, 6	cfv 5258	. . . 4 class (0g‘𝑟)
8	5, 7	wne 2367	. . 3 wff (1r‘𝑟) ≠ (0g‘𝑟)
9		crg 13552	. . 3 class Ring
10	8, 2, 9	crab 2479	. 2 class {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
11	1, 10	wceq 1364	1 wff NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}

This definition is referenced by:  isnzr  13737  nzrring  13739