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 = ( 0g𝑅 )
Assertion 0ring ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 𝐵 = { 0 } )

Proof

Step Hyp Ref Expression
1 0ring.b 𝐵 = ( Base ‘ 𝑅 )
2 0ring.0 0 = ( 0g𝑅 )
3 1 2 ring0cl ( 𝑅 ∈ Ring → 0𝐵 )
4 1 fvexi 𝐵 ∈ V
5 hashen1 ( 𝐵 ∈ V → ( ( ♯ ‘ 𝐵 ) = 1 ↔ 𝐵 ≈ 1o ) )
6 4 5 ax-mp ( ( ♯ ‘ 𝐵 ) = 1 ↔ 𝐵 ≈ 1o )
7 en1eqsn ( ( 0𝐵𝐵 ≈ 1o ) → 𝐵 = { 0 } )
8 7 ex ( 0𝐵 → ( 𝐵 ≈ 1o𝐵 = { 0 } ) )
9 6 8 syl5bi ( 0𝐵 → ( ( ♯ ‘ 𝐵 ) = 1 → 𝐵 = { 0 } ) )
10 3 9 syl ( 𝑅 ∈ Ring → ( ( ♯ ‘ 𝐵 ) = 1 → 𝐵 = { 0 } ) )
11 10 imp ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 𝐵 = { 0 } )