Theorem wunsuc

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

	Ref	Expression
Hypotheses	wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	wunsuc	⊢ ( 𝜑 → suc 𝐴 ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3		df-suc	⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
4	1 2	wunsn	⊢ ( 𝜑 → { 𝐴 } ∈ 𝑈 )
5	1 2 4	wunun	⊢ ( 𝜑 → ( 𝐴 ∪ { 𝐴 } ) ∈ 𝑈 )
6	3 5	eqeltrid	⊢ ( 𝜑 → suc 𝐴 ∈ 𝑈 )