Theorem uniex2

Description: The Axiom of Union using the standard abbreviation for union. Given any set x , its union y exists. (Contributed by NM, 4-Jun-2006) (Proof shortened by BJ, 14-Jul-2026)

	Ref	Expression
Assertion	uniex2	⊢ ∃ 𝑦 𝑦 = ∪ 𝑥

Proof

Step	Hyp	Ref	Expression
1		axun2	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) )
2		eluni	⊢ ( 𝑧 ∈ ∪ 𝑥 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) )
3	2	bibi2i	⊢ ( ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) ↔ ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) )
4	3	albii	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) )
5	4	exbii	⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) )
6	1 5	mpbir	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 )
7		dfcleq	⊢ ( 𝑦 = ∪ 𝑥 ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) )
8	7	biimpri	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) → 𝑦 = ∪ 𝑥 )
9	6 8	eximii	⊢ ∃ 𝑦 𝑦 = ∪ 𝑥