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Theorem neik0imk0p 39167
 Description: Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom imples 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 4150 . 2 (𝐵 ∈ (𝑁𝑥) → (𝑁𝑥) ≠ ∅)
21ralimi 3161 1 (∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  ∅c0 4144  ‘cfv 6123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-v 3416  df-dif 3801  df-nul 4145 This theorem is referenced by: (None)
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