Theorem 0cld

Description: The empty set is closed. Part of Theorem 6.1(1) of Munkres p. 93. (Contributed by NM, 4-Oct-2006)

	Ref	Expression
Assertion	0cld	⊢ ( 𝐽 ∈ Top → ∅ ∈ ( Clsd ‘ 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		dif0	⊢ ( ∪ 𝐽 ∖ ∅ ) = ∪ 𝐽
2	1	topopn	⊢ ( 𝐽 ∈ Top → ( ∪ 𝐽 ∖ ∅ ) ∈ 𝐽 )
3		0ss	⊢ ∅ ⊆ ∪ 𝐽
4		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
5	4	iscld2	⊢ ( ( 𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽 ) → ( ∅ ∈ ( Clsd ‘ 𝐽 ) ↔ ( ∪ 𝐽 ∖ ∅ ) ∈ 𝐽 ) )
6	3 5	mpan2	⊢ ( 𝐽 ∈ Top → ( ∅ ∈ ( Clsd ‘ 𝐽 ) ↔ ( ∪ 𝐽 ∖ ∅ ) ∈ 𝐽 ) )
7	2 6	mpbird	⊢ ( 𝐽 ∈ Top → ∅ ∈ ( Clsd ‘ 𝐽 ) )