| 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 22905 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∩ cin 3950 Topctop 22899 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 df-top 22900 |
| This theorem is referenced by: (None) |
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