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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 imples 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 3921 | . 2 ⊢ (𝐵 ∈ (𝑁‘𝑥) → (𝑁‘𝑥) ≠ ∅) | |
2 | 1 | ralimi 2952 | 1 ⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-nul 3916 |
This theorem is referenced by: (None) |
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