Theorem intssuni

Description: The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006)

	Ref	Expression
Assertion	intssuni	⊢ ( 𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		r19.2z	⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ) → ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 )
2	1	ex	⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ) )
3		vex	⊢ 𝑥 ∈ V
4	3	elint2	⊢ ( 𝑥 ∈ ∩ 𝐴 ↔ ∀ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 )
5		eluni2	⊢ ( 𝑥 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 )
6	2 4 5	3imtr4g	⊢ ( 𝐴 ≠ ∅ → ( 𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴 ) )
7	6	ssrdv	⊢ ( 𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴 )