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Theorem wunin 9108
Description: A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1  |-  ( ph  ->  U  e. WUni )
wununi.2  |-  ( ph  ->  A  e.  U )
Assertion
Ref Expression
wunin  |-  ( ph  ->  ( A  i^i  B
)  e.  U )

Proof of Theorem wunin
StepHypRef Expression
1 wununi.1 . 2  |-  ( ph  ->  U  e. WUni )
2 wununi.2 . 2  |-  ( ph  ->  A  e.  U )
3 inss1 3714 . . 3  |-  ( A  i^i  B )  C_  A
43a1i 11 . 2  |-  ( ph  ->  ( A  i^i  B
)  C_  A )
51, 2, 4wunss 9107 1  |-  ( ph  ->  ( A  i^i  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    i^i cin 3470    C_ wss 3471  WUnicwun 9095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017  df-uni 4252  df-tr 4551  df-wun 9097
This theorem is referenced by:  wunress  14710
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