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Metamath Proof Explorer

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 sepexi ⊢ ∃ 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