Definition df-nzr 13277

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 e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }

Detailed syntax breakdown of Definition df-nzr

Step	Hyp	Ref	Expression
1		cnzr 13276	. 2 class NzRing
2		vr	. . . . . 6 setvar r
3	2	cv 1352	. . . . 5 class r
4		cur 13095	. . . . 5 class 1r
5	3, 4	cfv 5216	. . . 4 class ( 1r ` r )
6		c0g 12695	. . . . 5 class 0g
7	3, 6	cfv 5216	. . . 4 class ( 0g ` r )
8	5, 7	wne 2347	. . 3 wff ( 1r ` r ) =/= ( 0g ` r )
9		crg 13132	. . 3 class Ring
10	8, 2, 9	crab 2459	. 2 class { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }
11	1, 10	wceq 1353	1 wff NzRing = { r e. Ring | ( 1r ` r ) =/= ( 0g ` r ) }

This definition is referenced by:  isnzr  13278  nzrring  13280