Metamath Proof Explorer


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

Proof

Step Hyp Ref Expression
1 0ringdif.b
 |-  B = ( Base ` R )
2 0ringdif.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. } )