Metamath Proof Explorer


Theorem restclssep

Description: Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024)

Ref Expression
Hypotheses restcls2.1 ( 𝜑𝐽 ∈ Top )
restcls2.2 ( 𝜑𝑋 = 𝐽 )
restcls2.3 ( 𝜑𝑌𝑋 )
restcls2.4 ( 𝜑𝐾 = ( 𝐽t 𝑌 ) )
restcls2.5 ( 𝜑𝑆 ∈ ( Clsd ‘ 𝐾 ) )
restclsseplem.6 ( 𝜑 → ( 𝑆𝑇 ) = ∅ )
restclssep.7 ( 𝜑𝑇 ∈ ( Clsd ‘ 𝐾 ) )
Assertion restclssep ( 𝜑 → ( ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) = ∅ ) )

Proof

Step Hyp Ref Expression
1 restcls2.1 ( 𝜑𝐽 ∈ Top )
2 restcls2.2 ( 𝜑𝑋 = 𝐽 )
3 restcls2.3 ( 𝜑𝑌𝑋 )
4 restcls2.4 ( 𝜑𝐾 = ( 𝐽t 𝑌 ) )
5 restcls2.5 ( 𝜑𝑆 ∈ ( Clsd ‘ 𝐾 ) )
6 restclsseplem.6 ( 𝜑 → ( 𝑆𝑇 ) = ∅ )
7 restclssep.7 ( 𝜑𝑇 ∈ ( Clsd ‘ 𝐾 ) )
8 incom ( ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ∩ 𝑆 ) = ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) )
9 incom ( 𝑆𝑇 ) = ( 𝑇𝑆 )
10 9 6 eqtr3id ( 𝜑 → ( 𝑇𝑆 ) = ∅ )
11 1 2 3 4 5 restcls2lem ( 𝜑𝑆𝑌 )
12 1 2 3 4 7 10 11 restclsseplem ( 𝜑 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ∩ 𝑆 ) = ∅ )
13 8 12 eqtr3id ( 𝜑 → ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) = ∅ )
14 1 2 3 4 7 restcls2lem ( 𝜑𝑇𝑌 )
15 1 2 3 4 5 6 14 restclsseplem ( 𝜑 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) = ∅ )
16 13 15 jca ( 𝜑 → ( ( 𝑆 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) = ∅ ) )