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 43363
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 4329 . 2 (𝐵 ∈ (𝑁𝑥) → (𝑁𝑥) ≠ ∅)
21ralimi 3077 1 (∀𝑥𝐵 𝐵 ∈ (𝑁𝑥) → ∀𝑥𝐵 (𝑁𝑥) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wne 2934  wral 3055  c0 4317  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-dif 3946  df-nul 4318
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator