Metamath Proof Explorer


Theorem wunint

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

Ref Expression
Hypotheses wununi.1
|- ( ph -> U e. WUni )
wununi.2
|- ( ph -> A e. U )
Assertion wunint
|- ( ( ph /\ A =/= (/) ) -> |^| A e. U )

Proof

Step Hyp Ref Expression
1 wununi.1
 |-  ( ph -> U e. WUni )
2 wununi.2
 |-  ( ph -> A e. U )
3 1 adantr
 |-  ( ( ph /\ A =/= (/) ) -> U e. WUni )
4 1 2 wununi
 |-  ( ph -> U. A e. U )
5 4 adantr
 |-  ( ( ph /\ A =/= (/) ) -> U. A e. U )
6 intssuni
 |-  ( A =/= (/) -> |^| A C_ U. A )
7 6 adantl
 |-  ( ( ph /\ A =/= (/) ) -> |^| A C_ U. A )
8 3 5 7 wunss
 |-  ( ( ph /\ A =/= (/) ) -> |^| A e. U )