Theorem inopnd 45725

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 23017	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)
5	1, 2, 3, 4	syl3anc 1394	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽)

Syntax hints:   → wi 4   ∈ wcel 2145   ∩ cin 3906  Topctop 23011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-in 3914  df-ss 3924  df-pw 4560  df-top 23012

This theorem is referenced by: (None)