Theorem wunop

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

	Ref	Expression
Hypotheses	wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
	wunop.2	$$⊢\phi \to A\in U$$
	wunop.3	$$⊢\phi \to B\in U$$
Assertion	wunop	$$⊢\phi \to ⟨A,B⟩\in U$$

Proof

Step	Hyp	Ref	Expression
1		wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wunop.2	$$⊢\phi \to A\in U$$
3		wunop.3	$$⊢\phi \to B\in U$$
4		dfopg	$$⊢\left(A\in U\wedge B\in U\right)\to ⟨A,B⟩=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
5	2 3 4	syl2anc	$$⊢\phi \to ⟨A,B⟩=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
6	1 2	wunsn	$$⊢\phi \to \left\{A\right\}\in U$$
7	1 2 3	wunpr	$$⊢\phi \to \left\{A,B\right\}\in U$$
8	1 6 7	wunpr	$$⊢\phi \to \left\{\left\{A\right\},\left\{A,B\right\}\right\}\in U$$
9	5 8	eqeltrd	$$⊢\phi \to ⟨A,B⟩\in U$$