Theorem wunsuc

Description: A weak universe is closed under successors. (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	wunsuc	$$⊢\phi \to \mathrm{suc}A\in U$$

Proof

Step	Hyp	Ref	Expression
1		wununi.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wununi.2	$$⊢\phi \to A\in U$$
3		df-suc	$$⊢\mathrm{suc}A=A\cup \left\{A\right\}$$
4	1 2	wunsn	$$⊢\phi \to \left\{A\right\}\in U$$
5	1 2 4	wunun	$$⊢\phi \to A\cup \left\{A\right\}\in U$$
6	3 5	eqeltrid	$$⊢\phi \to \mathrm{suc}A\in U$$