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	⊢ ( ∀ 𝑥 ∈ 𝐵 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝐵 ( 𝑁 ‘ 𝑥 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) → ( 𝑁 ‘ 𝑥 ) ≠ ∅ )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐵 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝐵 ( 𝑁 ‘ 𝑥 ) ≠ ∅ )