Theorem wunun

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

	Ref	Expression
Hypotheses	wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	wunpr.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
Assertion	wunun	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3		wunpr.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
4		uniprg	⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈 ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) )
5	2 3 4	syl2anc	⊢ ( 𝜑 → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) )
6	1 2 3	wunpr	⊢ ( 𝜑 → { 𝐴 , 𝐵 } ∈ 𝑈 )
7	1 6	wununi	⊢ ( 𝜑 → ∪ { 𝐴 , 𝐵 } ∈ 𝑈 )
8	5 7	eqeltrrd	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ 𝑈 )