Theorem neik0imk0p

Description: Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom implies KA' when considering the convergents. (Contributed by RP, 28-Jun-2021)

	Ref	Expression
Assertion	neik0imk0p	$$⊢\forall x\in BB\in N\left(x\right)\to \forall x\in BN\left(x\right)\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ne0i	$$⊢B\in N\left(x\right)\to N\left(x\right)\ne \varnothing$$
2	1	ralimi	$$⊢\forall x\in BB\in N\left(x\right)\to \forall x\in BN\left(x\right)\ne \varnothing$$