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Mirrors > Home > MPE Home > Th. List > Mathboxes > neik0imk0p | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
neik0imk0p | ⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4293 | . 2 ⊢ (𝐵 ∈ (𝑁‘𝑥) → (𝑁‘𝑥) ≠ ∅) | |
2 | 1 | ralimi 3159 | 1 ⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ∅c0 4284 ‘cfv 6348 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-ne 3016 df-ral 3142 df-dif 3932 df-nul 4285 |
This theorem is referenced by: (None) |
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