Metamath Proof Explorer

Theorem unexg

Description: A union of two sets is a set. Corollary 5.8 of TakeutiZaring p. 16. (Contributed by NM, 18-Sep-2006)

Ref Expression
Assertion unexg ⊢ A ∈ V ∧ B ∈ W → A ∪ B ∈ V

Proof

Step Hyp Ref Expression
1 elex ⊢ A ∈ V → A ∈ V
2 elex ⊢ B ∈ W → B ∈ V
3 unexb ⊢ A ∈ V ∧ B ∈ V ↔ A ∪ B ∈ V
4 3 biimpi ⊢ A ∈ V ∧ B ∈ V → A ∪ B ∈ V
5 1 2 4 syl2an ⊢ A ∈ V ∧ B ∈ W → A ∪ B ∈ V