Theorem neik0imk0p 41646

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 4268	. 2 ⊢ (𝐵 ∈ (𝑁‘𝑥) → (𝑁‘𝑥) ≠ ∅)
2	1	ralimi 3087	1 ⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∅c0 4256  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)