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	$$⊢B={\mathrm{Base}}_R$$
	rng1zr.p	$$⊢\underset{\dot{}}{+}={+}_R$$
	rng1zr.t	$$⊢\underset{\dot{}}{\ast }={\cdot }_R$$
Assertion	rngen1zr	$$⊢\left(\left(R\in \mathrm{Rng}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B\approx 1_{𝑜}\leftrightarrow \left(\underset{\dot{}}{+}=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\wedge \underset{\dot{}}{\ast }=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		rng1zr.b	$$⊢B={\mathrm{Base}}_R$$
2		rng1zr.p	$$⊢\underset{\dot{}}{+}={+}_R$$
3		rng1zr.t	$$⊢\underset{\dot{}}{\ast }={\cdot }_R$$
4		en1eqsnbi	$$⊢Z\in B\to \left(B\approx 1_{𝑜}\leftrightarrow B=\left\{Z\right\}\right)$$
5	4	adantl	$$⊢\left(\left(R\in \mathrm{Rng}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B\approx 1_{𝑜}\leftrightarrow B=\left\{Z\right\}\right)$$
6	1 2 3	rng1zr	$$⊢\left(\left(R\in \mathrm{Rng}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B=\left\{Z\right\}\leftrightarrow \left(\underset{\dot{}}{+}=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\wedge \underset{\dot{}}{\ast }=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\right)\right)$$
7	5 6	bitrd	$$⊢\left(\left(R\in \mathrm{Rng}\wedge \underset{\dot{}}{+}Fn\left(B\times B\right)\wedge \underset{\dot{}}{\ast }Fn\left(B\times B\right)\right)\wedge Z\in B\right)\to \left(B\approx 1_{𝑜}\leftrightarrow \left(\underset{\dot{}}{+}=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\wedge \underset{\dot{}}{\ast }=\left\{⟨⟨Z,Z⟩,Z⟩\right\}\right)\right)$$