Theorem wunss

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

	Ref	Expression
Hypotheses	wununi.1	$$⊢\phi \to U\in \mathrm{WUni}$$
	wununi.2	$$⊢\phi \to A\in U$$
	wunss.3	$$⊢\phi \to B\subseteq A$$
Assertion	wunss	$$⊢\phi \to B\in U$$

Proof

Step	Hyp	Ref	Expression
1		wununi.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wununi.2	$$⊢\phi \to A\in U$$
3		wunss.3	$$⊢\phi \to B\subseteq A$$
4	1 2	wunpw	$$⊢\phi \to 𝒫A\in U$$
5	1 4	wunelss	$$⊢\phi \to 𝒫A\subseteq U$$
6	2 3	sselpwd	$$⊢\phi \to B\in 𝒫A$$
7	5 6	sseldd	$$⊢\phi \to B\in U$$