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 e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. 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 e. B -> ( B ~~ 1o <-> B = { Z } ) )
5	4	adantl	|- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> B = { Z } ) )
6	1 2 3	rng1zr	|- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
7	5 6	bitrd	|- ( ( ( R e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )