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Theorem wunun 9142
Description: A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1  |-  ( ph  ->  U  e. WUni )
wununi.2  |-  ( ph  ->  A  e.  U )
wunpr.3  |-  ( ph  ->  B  e.  U )
Assertion
Ref Expression
wunun  |-  ( ph  ->  ( A  u.  B
)  e.  U )

Proof of Theorem wunun
StepHypRef Expression
1 wununi.2 . . 3  |-  ( ph  ->  A  e.  U )
2 wunpr.3 . . 3  |-  ( ph  ->  B  e.  U )
3 uniprg 4233 . . 3  |-  ( ( A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B )
)
41, 2, 3syl2anc 665 . 2  |-  ( ph  ->  U. { A ,  B }  =  ( A  u.  B )
)
5 wununi.1 . . 3  |-  ( ph  ->  U  e. WUni )
65, 1, 2wunpr 9141 . . 3  |-  ( ph  ->  { A ,  B }  e.  U )
75, 6wununi 9138 . 2  |-  ( ph  ->  U. { A ,  B }  e.  U
)
84, 7eqeltrrd 2508 1  |-  ( ph  ->  ( A  u.  B
)  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    u. cun 3434   {cpr 4000   U.cuni 4219  WUnicwun 9132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-un 3441  df-in 3443  df-ss 3450  df-sn 3999  df-pr 4001  df-uni 4220  df-tr 4519  df-wun 9134
This theorem is referenced by:  wuntp  9143  wunsuc  9149  wunfi  9153  wunxp  9156  wuntpos  9166  wunsets  15149  catcoppccl  16002
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