Theorem 0ring1eq0

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

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

Proof

Step	Hyp	Ref	Expression
1		0ringdif.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		0ringdif.0	⊢ 0 = ( 0g ‘ 𝑅 )
3		0ring1eq0.1	⊢ 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 )