Theorem zfcndun

Description: Axiom of Union ax-un , reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Aug-2003) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

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