Theorem 0ringbas

Description: The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020)

	Ref	Expression
Hypotheses	0ring.b	|- B = ( Base ` R )
	0ring.0	|- .0. = ( 0g ` R )
Assertion	0ringbas	|- ( R e. ( Ring \ NzRing ) -> B = { .0. } )

Proof

Step	Hyp	Ref	Expression
1		0ring.b	|- B = ( Base ` R )
2		0ring.0	|- .0. = ( 0g ` R )
3	1 2	0ringdif	|- ( R e. ( Ring \ NzRing ) <-> ( R e. Ring /\ B = { .0. } ) )
4	3	simprbi	|- ( R e. ( Ring \ NzRing ) -> B = { .0. } )