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Metamath Proof Explorer

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 )