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Metamath Proof Explorer

Theorem 0ring1eq0

Description: In a zero ring, a ring which is not a nonzero ring, the ring unity equals the zero element. (Contributed by AV, 17-Apr-2020)

Ref Expression
Hypotheses 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion 0ring1eq0 ( 𝑅 ∈ ( Ring ∖ NzRing ) → 1 = 0 )

Proof

Step Hyp Ref Expression
1 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
3 0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
4 eldif ⊢ ( 𝑅 ∈ ( Ring ∖ NzRing ) ↔ ( 𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing ) )
5 0ringnnzr ⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ↔ ¬ 𝑅 ∈ NzRing ) )
6 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
7 6 2 3 0ring01eq ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ) → 0 = 1 )
8 7 eqcomd ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ) → 1 = 0 )
9 8 ex ⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 → 1 = 0 ) )
10 5 9 sylbird ⊢ ( 𝑅 ∈ Ring → ( ¬ 𝑅 ∈ NzRing → 1 = 0 ) )
11 10 imp ⊢ ( ( 𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing ) → 1 = 0 )
12 4 11 sylbi ⊢ ( 𝑅 ∈ ( Ring ∖ NzRing ) → 1 = 0 )