Theorem wunxp

Description: A weak universe is closed under cartesian products. (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	wunxp	|- ( ph -> ( A X. 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	1 2 3	wunun	|- ( ph -> ( A u. B ) e. U )
5	1 4	wunpw	|- ( ph -> ~P ( A u. B ) e. U )
6	1 5	wunpw	|- ( ph -> ~P ~P ( A u. B ) e. U )
7		xpsspw	|- ( A X. B ) C_ ~P ~P ( A u. B )
8	7	a1i	|- ( ph -> ( A X. B ) C_ ~P ~P ( A u. B ) )
9	1 6 8	wunss	|- ( ph -> ( A X. B ) e. U )