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Metamath Proof Explorer

Theorem shincli

Description: Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses shincl.1 ⊢ A ∈ S ℋ
shincl.2 ⊢ B ∈ S ℋ
Assertion shincli ⊢ A ∩ B ∈ S ℋ

Proof

Step Hyp Ref Expression
1 shincl.1 ⊢ A ∈ S ℋ
2 shincl.2 ⊢ B ∈ S ℋ
3 1 elexi ⊢ A ∈ V
4 2 elexi ⊢ B ∈ V
5 3 4 intpr ⊢ ⋂ A B = A ∩ B
6 1 2 pm3.2i ⊢ A ∈ S ℋ ∧ B ∈ S ℋ
7 3 4 prss ⊢ A ∈ S ℋ ∧ B ∈ S ℋ ↔ A B ⊆ S ℋ
8 6 7 mpbi ⊢ A B ⊆ S ℋ
9 3 prnz ⊢ A B ≠ ∅
10 8 9 pm3.2i ⊢ A B ⊆ S ℋ ∧ A B ≠ ∅
11 10 shintcli ⊢ ⋂ A B ∈ S ℋ
12 5 11 eqeltrri ⊢ A ∩ B ∈ S ℋ