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Theorem neik0imk0p 40386
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 (∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)

Proof of Theorem neik0imk0p
StepHypRef Expression
1 ne0i 4299 . 2 (𝐵 ∈ (𝑁𝑥) → (𝑁𝑥) ≠ ∅)
21ralimi 3160 1 (∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wne 3016  wral 3138  c0 4290  cfv 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-ral 3143  df-dif 3938  df-nul 4291
This theorem is referenced by: (None)
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