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Metamath Proof Explorer

Theorem wunun

Description: A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wununi.1 ⊢ φ → U ∈ WUni
wununi.2 ⊢ φ → A ∈ U
wunpr.3 ⊢ φ → B ∈ U
Assertion wunun ⊢ φ → A ∪ B ∈ U

Proof

Step Hyp Ref Expression
1 wununi.1 ⊢ φ → U ∈ WUni
2 wununi.2 ⊢ φ → A ∈ U
3 wunpr.3 ⊢ φ → B ∈ U
4 uniprg ⊢ A ∈ U ∧ B ∈ U → ⋃ A B = A ∪ B
5 2 3 4 syl2anc ⊢ φ → ⋃ A B = A ∪ B
6 1 2 3 wunpr ⊢ φ → A B ∈ U
7 1 6 wununi ⊢ φ → ⋃ A B ∈ U
8 5 7 eqeltrrd ⊢ φ → A ∪ B ∈ U