Theorem inopnd 45154

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	⊢ (𝜑 → 𝐽 ∈ Top)
inopnd.2	⊢ (𝜑 → 𝐴 ∈ 𝐽)
inopnd.3	⊢ (𝜑 → 𝐵 ∈ 𝐽)

Assertion

Ref	Expression
inopnd	⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽)

Proof of Theorem inopnd

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)

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)