Theorem uniel

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

	Ref	Expression
Assertion	uniel	⊢ ( ∪ 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		clabel	⊢ ( { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 } ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ) ) )
2		dfuni2	⊢ ∪ 𝐴 = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 }
3	2	eleq1i	⊢ ( ∪ 𝐴 ∈ 𝐵 ↔ { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 } ∈ 𝐵 )
4		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ) ) )
5	1 3 4	3bitr4i	⊢ ( ∪ 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ) )