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Theorem inopnd 45091
Description: The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Hypotheses
Ref Expression
inopnd.1 (𝜑𝐽 ∈ Top)
inopnd.2 (𝜑𝐴𝐽)
inopnd.3 (𝜑𝐵𝐽)
Assertion
Ref Expression
inopnd (𝜑 → (𝐴𝐵) ∈ 𝐽)

Proof of Theorem inopnd
StepHypRef Expression
1 inopnd.1 . 2 (𝜑𝐽 ∈ Top)
2 inopnd.2 . 2 (𝜑𝐴𝐽)
3 inopnd.3 . 2 (𝜑𝐵𝐽)
4 inopn 22920 . 2 ((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
51, 2, 3, 4syl3anc 1370 1 (𝜑 → (𝐴𝐵) ∈ 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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