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Metamath Proof Explorer

Theorem 0ring01eqbi

Description: In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010) (Revised by AV, 23-Jan-2020)

Ref Expression
Hypotheses 0ring.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
0ring.0 ⊢ 0 = ( 0g ‘ 𝑅 )
0ring01eq.1 ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion 0ring01eqbi ( 𝑅 ∈ Ring → ( 𝐵 ≈ 1o ↔ 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 1 fvexi ⊢ 𝐵 ∈ V
5 hashen1 ⊢ ( 𝐵 ∈ V → ( ( ♯ ‘ 𝐵 ) = 1 ↔ 𝐵 ≈ 1o ) )
6 4 5 mp1i ⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ 𝐵 ) = 1 ↔ 𝐵 ≈ 1o ) )
7 1 2 3 0ring01eq ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 0 = 1 )
8 7 eqcomd ⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 1 = 0 )
9 8 ex ⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ 𝐵 ) = 1 → 1 = 0 ) )
10 eqcom ⊢ ( 1 = 0 ↔ 0 = 1 )
11 1 2 3 01eq0ring ⊢ ( ( 𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 } )
12 fveq2 ⊢ ( 𝐵 = { 0 } → ( ♯ ‘ 𝐵 ) = ( ♯ ‘ { 0 } ) )
13 2 fvexi ⊢ 0 ∈ V
14 hashsng ⊢ ( 0 ∈ V → ( ♯ ‘ { 0 } ) = 1 )
15 13 14 mp1i ⊢ ( 𝐵 = { 0 } → ( ♯ ‘ { 0 } ) = 1 )
16 12 15 eqtrd ⊢ ( 𝐵 = { 0 } → ( ♯ ‘ 𝐵 ) = 1 )
17 11 16 syl ⊢ ( ( 𝑅 ∈ Ring ∧ 0 = 1 ) → ( ♯ ‘ 𝐵 ) = 1 )
18 17 ex ⊢ ( 𝑅 ∈ Ring → ( 0 = 1 → ( ♯ ‘ 𝐵 ) = 1 ) )
19 10 18 biimtrid ⊢ ( 𝑅 ∈ Ring → ( 1 = 0 → ( ♯ ‘ 𝐵 ) = 1 ) )
20 9 19 impbid ⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ 𝐵 ) = 1 ↔ 1 = 0 ) )
21 6 20 bitr3d ⊢ ( 𝑅 ∈ Ring → ( 𝐵 ≈ 1o ↔ 1 = 0 ) )