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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 22745 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∩ cin 3940 Topctop 22739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-in 3948 df-ss 3958 df-pw 4597 df-top 22740 |
This theorem is referenced by: (None) |
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