Theorem chincli

Description: Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
Assertion	chincli	⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3	1	elexi	⊢ 𝐴 ∈ V
4	2	elexi	⊢ 𝐵 ∈ V
5	3 4	intpr	⊢ ∩ { 𝐴 , 𝐵 } = ( 𝐴 ∩ 𝐵 )
6	1 2	pm3.2i	⊢ ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )
7	3 4	prss	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ↔ { 𝐴 , 𝐵 } ⊆ Cℋ )
8	6 7	mpbi	⊢ { 𝐴 , 𝐵 } ⊆ Cℋ
9	3	prnz	⊢ { 𝐴 , 𝐵 } ≠ ∅
10	8 9	pm3.2i	⊢ ( { 𝐴 , 𝐵 } ⊆ Cℋ ∧ { 𝐴 , 𝐵 } ≠ ∅ )
11	10	chintcli	⊢ ∩ { 𝐴 , 𝐵 } ∈ Cℋ
12	5 11	eqeltrri	⊢ ( 𝐴 ∩ 𝐵 ) ∈ Cℋ