Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neik0imk0p Structured version   Visualization version   GIF version

Theorem neik0imk0p 44049
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 4341 . 2 (𝐵 ∈ (𝑁𝑥) → (𝑁𝑥) ≠ ∅)
21ralimi 3083 1 (∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wne 2940  wral 3061  c0 4333  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-dif 3954  df-nul 4334
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator