This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem wunss

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

Ref Expression
Hypotheses wununi.1 ⊢ φ → U ∈ WUni
wununi.2 ⊢ φ → A ∈ U
wunss.3 ⊢ φ → B ⊆ A
Assertion wunss ⊢ φ → B ∈ U

Proof

Step Hyp Ref Expression
1 wununi.1 ⊢ φ → U ∈ WUni
2 wununi.2 ⊢ φ → A ∈ U
3 wunss.3 ⊢ φ → B ⊆ A
4 1 2 wunpw ⊢ φ → 𝒫 A ∈ U
5 1 4 wunelss ⊢ φ → 𝒫 A ⊆ U
6 2 3 sselpwd ⊢ φ → B ∈ 𝒫 A
7 5 6 sseldd ⊢ φ → B ∈ U