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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inopnd | Structured version Visualization version GIF version |
Description: The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
Ref | Expression |
---|---|
inopnd.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
inopnd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
inopnd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝐽) |
Ref | Expression |
---|---|
inopnd | ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inopnd.1 | . 2 ⊢ (𝜑 → 𝐽 ∈ Top) | |
2 | inopnd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
3 | inopnd.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐽) | |
4 | inopn 22920 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3961 Topctop 22914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-in 3969 df-ss 3979 df-pw 4606 df-top 22915 |
This theorem is referenced by: (None) |
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