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Theorem wunin 8901
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 3591 . . 3  |-  ( A  i^i  B )  C_  A
43a1i 11 . 2  |-  ( ph  ->  ( A  i^i  B
)  C_  A )
51, 2, 4wunss 8900 1  |-  ( ph  ->  ( A  i^i  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    i^i cin 3348    C_ wss 3349  WUnicwun 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-v 2995  df-in 3356  df-ss 3363  df-pw 3883  df-uni 4113  df-tr 4407  df-wun 8890
This theorem is referenced by:  wunress  14258
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