Theorem wununi

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

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

Proof

Step	Hyp	Ref	Expression
1		wununi.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wununi.2	$$⊢\phi \to A\in U$$
3		unieq	$$⊢x=A\to \bigcup x=\bigcup A$$
4	3	eleq1d	$$⊢x=A\to \left(\bigcup x\in U\leftrightarrow \bigcup A\in U\right)$$
5		iswun	$$⊢U\in \mathrm{WUni}\to \left(U\in \mathrm{WUni}\leftrightarrow \left(\mathrm{Tr}U\wedge U\ne \varnothing \wedge \forall x\in U\left(\bigcup x\in U\wedge 𝒫x\in U\wedge \forall y\in U\left\{x,y\right\}\in U\right)\right)\right)$$
6	5	ibi	$$⊢U\in \mathrm{WUni}\to \left(\mathrm{Tr}U\wedge U\ne \varnothing \wedge \forall x\in U\left(\bigcup x\in U\wedge 𝒫x\in U\wedge \forall y\in U\left\{x,y\right\}\in U\right)\right)$$
7	6	simp3d	$$⊢U\in \mathrm{WUni}\to \forall x\in U\left(\bigcup x\in U\wedge 𝒫x\in U\wedge \forall y\in U\left\{x,y\right\}\in U\right)$$
8		simp1	$$⊢\left(\bigcup x\in U\wedge 𝒫x\in U\wedge \forall y\in U\left\{x,y\right\}\in U\right)\to \bigcup x\in U$$
9	8	ralimi	$$⊢\forall x\in U\left(\bigcup x\in U\wedge 𝒫x\in U\wedge \forall y\in U\left\{x,y\right\}\in U\right)\to \forall x\in U\bigcup x\in U$$
10	1 7 9	3syl	$$⊢\phi \to \forall x\in U\bigcup x\in U$$
11	4 10 2	rspcdva	$$⊢\phi \to \bigcup A\in U$$