Theorem eltopss

Description: A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006)

	Ref	Expression
Hypothesis	1open.1	⊢ 𝑋 = ∪ 𝐽
Assertion	eltopss	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		1open.1	⊢ 𝑋 = ∪ 𝐽
2		elssuni	⊢ ( 𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽 )
3	2 1	sseqtrrdi	⊢ ( 𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋 )
4	3	adantl	⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )