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Theorem wunss 8993
Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1  |-  ( ph  ->  U  e. WUni )
wununi.2  |-  ( ph  ->  A  e.  U )
wunss.3  |-  ( ph  ->  B  C_  A )
Assertion
Ref Expression
wunss  |-  ( ph  ->  B  e.  U )

Proof of Theorem wunss
StepHypRef Expression
1 wununi.1 . . 3  |-  ( ph  ->  U  e. WUni )
2 wununi.2 . . . 4  |-  ( ph  ->  A  e.  U )
31, 2wunpw 8988 . . 3  |-  ( ph  ->  ~P A  e.  U
)
41, 3wunelss 8989 . 2  |-  ( ph  ->  ~P A  C_  U
)
5 wunss.3 . . 3  |-  ( ph  ->  B  C_  A )
6 elpw2g 4566 . . . 4  |-  ( A  e.  U  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
72, 6syl 16 . . 3  |-  ( ph  ->  ( B  e.  ~P A 
<->  B  C_  A )
)
85, 7mpbird 232 . 2  |-  ( ph  ->  B  e.  ~P A
)
94, 8sseldd 3468 1  |-  ( ph  ->  B  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1758    C_ wss 3439   ~Pcpw 3971  WUnicwun 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-in 3446  df-ss 3453  df-pw 3973  df-uni 4203  df-tr 4497  df-wun 8983
This theorem is referenced by:  wunin  8994  wundif  8995  wunint  8996  wun0  8999  wunom  9001  wunxp  9005  wunpm  9006  wunmap  9007  wundm  9009  wunrn  9010  wuncnv  9011  wunres  9012  wunfv  9013  wunco  9014  wuntpos  9015  wuncn  9451  wunndx  14311  wunstr  14314  wunfunc  14931
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