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)

	Ref	Expression
Assertion	uniex2	⊢ ∃ 𝑦 𝑦 = ∪ 𝑥

Proof

Step	Hyp	Ref	Expression
1		ax-un	⊢ ∃ 𝑦 ∀ 𝑧 ( ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 )
2		eluni	⊢ ( 𝑧 ∈ ∪ 𝑥 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) )
3	2	imbi1i	⊢ ( ( 𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ( ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) )
4	3	albii	⊢ ( ∀ 𝑧 ( 𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ∀ 𝑧 ( ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) )
5	4	exbii	⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦 ) ↔ ∃ 𝑦 ∀ 𝑧 ( ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → 𝑧 ∈ 𝑦 ) )
6	1 5	mpbir	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦 )
7	6	bm1.3ii	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 )
8		dfcleq	⊢ ( 𝑦 = ∪ 𝑥 ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) )
9	8	exbii	⊢ ( ∃ 𝑦 𝑦 = ∪ 𝑥 ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥 ) )
10	7 9	mpbir	⊢ ∃ 𝑦 𝑦 = ∪ 𝑥