Theorem neik0imk0p 44193

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	⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Proof of Theorem neik0imk0p

Step	Hyp	Ref	Expression
1		ne0i 4290	. 2 ⊢ (𝐵 ∈ (𝑁‘𝑥) → (𝑁‘𝑥) ≠ ∅)
2	1	ralimi 3070	1 ⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∅c0 4282  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-dif 3901  df-nul 4283

This theorem is referenced by: (None)