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Theorem neik0imk0p 43998
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 4364 . 2 (𝐵 ∈ (𝑁𝑥) → (𝑁𝑥) ≠ ∅)
21ralimi 3089 1 (∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wne 2946  wral 3067  c0 4352  cfv 6573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-dif 3979  df-nul 4353
This theorem is referenced by: (None)
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