Theorem wunop

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

	Ref	Expression
Hypotheses	wun0.1	|- ( ph -> U e. WUni )
	wunop.2	|- ( ph -> A e. U )
	wunop.3	|- ( ph -> B e. U )
Assertion	wunop	|- ( ph -> <. A , B >. e. U )

Proof

Step	Hyp	Ref	Expression
1		wun0.1	|- ( ph -> U e. WUni )
2		wunop.2	|- ( ph -> A e. U )
3		wunop.3	|- ( ph -> B e. U )
4		dfopg	|- ( ( A e. U /\ B e. U ) -> <. A , B >. = { { A } , { A , B } } )
5	2 3 4	syl2anc	|- ( ph -> <. A , B >. = { { A } , { A , B } } )
6	1 2	wunsn	|- ( ph -> { A } e. U )
7	1 2 3	wunpr	|- ( ph -> { A , B } e. U )
8	1 6 7	wunpr	|- ( ph -> { { A } , { A , B } } e. U )
9	5 8	eqeltrd	|- ( ph -> <. A , B >. e. U )