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Metamath Proof Explorer

Theorem 0ring

Description: If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019)

Ref Expression
Hypotheses 0ring.b 
|- B = ( Base ` R )
0ring.0 
|- .0. = ( 0g ` R )
Assertion 0ring 
|- ( ( R e. Ring /\ ( # ` B ) = 1 ) -> B = { .0. } )

Proof

Step Hyp Ref Expression
1 0ring.b 
 |-  B = ( Base ` R )
2 0ring.0 
 |-  .0. = ( 0g ` R )
3 1 2 ring0cl 
 |-  ( R e. Ring -> .0. e. B )
4 1 fvexi 
 |-  B e. _V
5 hashen1 
 |-  ( B e. _V -> ( ( # ` B ) = 1 <-> B ~~ 1o ) )
6 4 5 ax-mp 
 |-  ( ( # ` B ) = 1 <-> B ~~ 1o )
7 en1eqsn 
 |-  ( ( .0. e. B /\ B ~~ 1o ) -> B = { .0. } )
8 7 ex 
 |-  ( .0. e. B -> ( B ~~ 1o -> B = { .0. } ) )
9 6 8 biimtrid 
 |-  ( .0. e. B -> ( ( # ` B ) = 1 -> B = { .0. } ) )
10 3 9 syl 
 |-  ( R e. Ring -> ( ( # ` B ) = 1 -> B = { .0. } ) )
11 10 imp 
 |-  ( ( R e. Ring /\ ( # ` B ) = 1 ) -> B = { .0. } )