Theorem ssunib

Description: Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025)

	Ref	Expression
Assertion	ssunib	⊢ ( 𝐴 ⊆ ∪ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		dfss3	⊢ ( 𝐴 ⊆ ∪ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 )
2		eluni2	⊢ ( 𝑥 ∈ ∪ 𝐵 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 )
4	1 3	bitri	⊢ ( 𝐴 ⊆ ∪ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 )