Metamath Proof Explorer


Theorem iscnrm4

Description: A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024)

Ref Expression
Assertion iscnrm4 ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛𝑚 ) = ∅ ) ) )

Proof

Step Hyp Ref Expression
1 iscnrm3 ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
2 id ( 𝐽 ∈ Top → 𝐽 ∈ Top )
3 2 sepnsepo ( 𝐽 ∈ Top → ( ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛𝑚 ) = ∅ ↔ ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) )
4 3 imbi2d ( 𝐽 ∈ Top → ( ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛𝑚 ) = ∅ ) ↔ ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
5 4 2ralbidv ( 𝐽 ∈ Top → ( ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛𝑚 ) = ∅ ) ↔ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
6 5 pm5.32i ( ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛𝑚 ) = ∅ ) ) ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
7 1 6 bitr4i ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑠 ) ∃ 𝑚 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑡 ) ( 𝑛𝑚 ) = ∅ ) ) )