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 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shincli	⊢ ( 𝐴 ∩ 𝐵 ) ∈ Sℋ

Proof

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