Theorem wunccl

Description: The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Assertion	wunccl	$$⊢A\in V\to \mathrm{wUniCl}\left(A\right)\in \mathrm{WUni}$$

Proof

Step	Hyp	Ref	Expression
1		wuncval	$$⊢A\in V\to \mathrm{wUniCl}\left(A\right)=\bigcap \left\{u\in \mathrm{WUni}|A\subseteq u\right\}$$
2		ssrab2	$$⊢\left\{u\in \mathrm{WUni}|A\subseteq u\right\}\subseteq \mathrm{WUni}$$
3		wunex	$$⊢A\in V\to \exists u\in \mathrm{WUni}A\subseteq u$$
4		rabn0	$$⊢\left\{u\in \mathrm{WUni}|A\subseteq u\right\}\ne \varnothing \leftrightarrow \exists u\in \mathrm{WUni}A\subseteq u$$
5	3 4	sylibr	$$⊢A\in V\to \left\{u\in \mathrm{WUni}|A\subseteq u\right\}\ne \varnothing$$
6		intwun	$$⊢\left(\left\{u\in \mathrm{WUni}|A\subseteq u\right\}\subseteq \mathrm{WUni}\wedge \left\{u\in \mathrm{WUni}|A\subseteq u\right\}\ne \varnothing \right)\to \bigcap \left\{u\in \mathrm{WUni}|A\subseteq u\right\}\in \mathrm{WUni}$$
7	2 5 6	sylancr	$$⊢A\in V\to \bigcap \left\{u\in \mathrm{WUni}|A\subseteq u\right\}\in \mathrm{WUni}$$
8	1 7	eqeltrd	$$⊢A\in V\to \mathrm{wUniCl}\left(A\right)\in \mathrm{WUni}$$