Theorem rngen1zr

Description: The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010) (Revised by AV, 18-Jun-2026)

	Ref	Expression
Hypotheses	rng1zr.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rng1zr.p	⊢ + = ( +g ‘ 𝑅 )
	rng1zr.t	⊢ ∗ = ( .r ‘ 𝑅 )
Assertion	rngen1zr	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 ≈ 1o ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		rng1zr.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rng1zr.p	⊢ + = ( +g ‘ 𝑅 )
3		rng1zr.t	⊢ ∗ = ( .r ‘ 𝑅 )
4		en1eqsnbi	⊢ ( 𝑍 ∈ 𝐵 → ( 𝐵 ≈ 1o ↔ 𝐵 = { 𝑍 } ) )
5	4	adantl	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 ≈ 1o ↔ 𝐵 = { 𝑍 } ) )
6	1 2 3	rng1zr	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )
7	5 6	bitrd	⊢ ( ( ( 𝑅 ∈ Rng ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 ≈ 1o ↔ ( + = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ∧ ∗ = { ⟨ ⟨ 𝑍 , 𝑍 ⟩ , 𝑍 ⟩ } ) ) )