Theorem neik0imk0p 45021

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.)

Assertion

Ref	Expression
neik0imk0p	⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Proof of Theorem neik0imk0p

Step	Hyp	Ref	Expression
1		ne0i 4287	. 2 ⊢ (𝐵 ∈ (𝑁‘𝑥) → (𝑁‘𝑥) ≠ ∅)
2	1	ralimi 3100	1 ⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∅c0 4279  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-dif 3902  df-nul 4280

This theorem is used by: (None)