Theorem unielid

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

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

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐴 ⊆ 𝐴
2		unielss	⊢ ( 𝐴 ⊆ 𝐴 → ( ∪ 𝐴 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ) )
3	1 2	ax-mp	⊢ ( ∪ 𝐴 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 )