Theorem zfun

Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003) Use ax-un instead. (New usage is discouraged.)

	Ref	Expression
Assertion	zfun	⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ax-un	⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑤 ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )
2		elequ2	⊢ ( 𝑤 = 𝑥 → ( 𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑥 ) )
3		elequ1	⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧 ) )
4	2 3	anbi12d	⊢ ( 𝑤 = 𝑥 → ( ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧 ) ↔ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) )
5	4	cbvexvw	⊢ ( ∃ 𝑤 ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧 ) ↔ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) )
6	5	imbi1i	⊢ ( ( ∃ 𝑤 ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
7	6	albii	⊢ ( ∀ 𝑦 ( ∃ 𝑤 ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑤 ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
9	1 8	mpbi	⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )