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Metamath Proof Explorer

Theorem eliunid

Description: Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Assertion eliunid ⊢ x ∈ A ∧ C ∈ B → C ∈ ⋃ x ∈ A B

Proof

Step Hyp Ref Expression
1 rspe ⊢ x ∈ A ∧ C ∈ B → ∃ x ∈ A C ∈ B
2 eliun ⊢ C ∈ ⋃ x ∈ A B ↔ ∃ x ∈ A C ∈ B
3 1 2 sylibr ⊢ x ∈ A ∧ C ∈ B → C ∈ ⋃ x ∈ A B