Theorem intss2

Description: A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021)

	Ref	Expression
Assertion	intss2	⊢ ( 𝐴 ⊆ 𝒫 𝑋 → ( 𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		sspwuni	⊢ ( 𝐴 ⊆ 𝒫 𝑋 ↔ ∪ 𝐴 ⊆ 𝑋 )
2	1	biimpi	⊢ ( 𝐴 ⊆ 𝒫 𝑋 → ∪ 𝐴 ⊆ 𝑋 )
3		intssuni	⊢ ( 𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴 )
4		sstr	⊢ ( ( ∩ 𝐴 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑋 ) → ∩ 𝐴 ⊆ 𝑋 )
5	4	expcom	⊢ ( ∪ 𝐴 ⊆ 𝑋 → ( ∩ 𝐴 ⊆ ∪ 𝐴 → ∩ 𝐴 ⊆ 𝑋 ) )
6	2 3 5	syl2im	⊢ ( 𝐴 ⊆ 𝒫 𝑋 → ( 𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋 ) )