Theorem wunun

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

	Ref	Expression
Hypotheses	wununi.1	|- ( ph -> U e. WUni )
	wununi.2	|- ( ph -> A e. U )
	wunpr.3	|- ( ph -> B e. U )
Assertion	wunun	|- ( ph -> ( A u. B ) e. U )

Proof

Step	Hyp	Ref	Expression
1		wununi.1	|- ( ph -> U e. WUni )
2		wununi.2	|- ( ph -> A e. U )
3		wunpr.3	|- ( ph -> B e. U )
4		uniprg	|- ( ( A e. U /\ B e. U ) -> U. { A , B } = ( A u. B ) )
5	2 3 4	syl2anc	|- ( ph -> U. { A , B } = ( A u. B ) )
6	1 2 3	wunpr	|- ( ph -> { A , B } e. U )
7	1 6	wununi	|- ( ph -> U. { A , B } e. U )
8	5 7	eqeltrrd	|- ( ph -> ( A u. B ) e. U )