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 ) |
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 ) |