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Metamath Proof Explorer

Theorem wunss

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

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

Proof

Step Hyp Ref Expression
1 wununi.1 
 |-  ( ph -> U e. WUni )
2 wununi.2 
 |-  ( ph -> A e. U )
3 wunss.3 
 |-  ( ph -> B C_ A )
4 1 2 wunpw 
 |-  ( ph -> ~P A e. U )
5 1 4 wunelss 
 |-  ( ph -> ~P A C_ U )
6 2 3 sselpwd 
 |-  ( ph -> B e. ~P A )
7 5 6 sseldd 
 |-  ( ph -> B e. U )