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Metamath Proof Explorer

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 )