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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 ∈ 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 ∈ Ring → 0 ∈ 𝐵 )
4 1 fvexi ⊢ 𝐵 ∈ V
5 hashen1 ⊢ ( 𝐵 ∈ V → ( ( ♯ ‘ 𝐵 ) = 1 ↔ 𝐵 ≈ 1o ) )
6 4 5 ax-mp ⊢ ( ( ♯ ‘ 𝐵 ) = 1 ↔ 𝐵 ≈ 1o )
7 en1eqsn ⊢ ( ( 0 ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → 𝐵 = { 0 } )
8 7 ex ⊢ ( 0 ∈ 𝐵 → ( 𝐵 ≈ 1o → 𝐵 = { 0 } ) )
9 6 8 biimtrid ⊢ ( 0 ∈ 𝐵 → ( ( ♯ ‘ 𝐵 ) = 1 → 𝐵 = { 0 } ) )
10 3 9 syl ⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ 𝐵 ) = 1 → 𝐵 = { 0 } ) )
11 10 imp ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 𝐵 = { 0 } )