Theorem inopnd 46437

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 23272	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)
5	1, 2, 3, 4	syl3anc 1399	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092  Topctop 23266

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562  df-top 23267

This theorem is used by: (None)