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Metamath Proof Explorer

Theorem 0ring01eq

Description: In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019)

Ref Expression
Hypotheses 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion 0ring01eq ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 0 = 1 )

Proof

Step Hyp Ref Expression
1 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
4 1 2 0ring ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 𝐵 = { 0 } )
5 1 3 ringidcl ⊢ ( 𝑅 ∈ Ring → 1 ∈ 𝐵 )
6 eleq2 ⊢ ( 𝐵 = { 0 } → ( 1 ∈ 𝐵 ↔ 1 ∈ { 0 } ) )
7 elsni ⊢ ( 1 ∈ { 0 } → 1 = 0 )
8 7 eqcomd ⊢ ( 1 ∈ { 0 } → 0 = 1 )
9 6 8 biimtrdi ⊢ ( 𝐵 = { 0 } → ( 1 ∈ 𝐵 → 0 = 1 ) )
10 5 9 syl5com ⊢ ( 𝑅 ∈ Ring → ( 𝐵 = { 0 } → 0 = 1 ) )
11 10 adantr ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → ( 𝐵 = { 0 } → 0 = 1 ) )
12 4 11 mpd ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 0 = 1 )