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Theorem gruun 9218
Description: A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruun  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  e.  U )

Proof of Theorem gruun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 uniiun 4301 . . 3  |-  U. { A ,  B }  =  U_ x  e.  { A ,  B }
x
2 uniprg 4182 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B )
)
323adant1 1027 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B ) )
41, 3syl5reqr 2501 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  =  U_ x  e.  { A ,  B }
x )
5 simp1 1009 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
6 grupr 9209 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 vex 3016 . . . . . . 7  |-  x  e. 
_V
87elpr 3954 . . . . . 6  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
9 eleq1a 2525 . . . . . . 7  |-  ( A  e.  U  ->  (
x  =  A  ->  x  e.  U )
)
10 eleq1a 2525 . . . . . . 7  |-  ( B  e.  U  ->  (
x  =  B  ->  x  e.  U )
)
119, 10jaao 516 . . . . . 6  |-  ( ( A  e.  U  /\  B  e.  U )  ->  ( ( x  =  A  \/  x  =  B )  ->  x  e.  U ) )
128, 11syl5bi 225 . . . . 5  |-  ( ( A  e.  U  /\  B  e.  U )  ->  ( x  e.  { A ,  B }  ->  x  e.  U ) )
1312ralrimiv 2789 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B }
x  e.  U )
14133adant1 1027 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B } x  e.  U
)
15 gruiun 9211 . . 3  |-  ( ( U  e.  Univ  /\  { A ,  B }  e.  U  /\  A. x  e.  { A ,  B } x  e.  U
)  ->  U_ x  e. 
{ A ,  B } x  e.  U
)
165, 6, 14, 15syl3anc 1271 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U_ x  e.  { A ,  B } x  e.  U
)
174, 16eqeltrd 2530 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375    /\ w3a 986    = wceq 1448    e. wcel 1891   A.wral 2737    u. cun 3370   {cpr 3938   U.cuni 4168   U_ciun 4248   Univcgru 9202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-fv 5569  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-map 7461  df-gru 9203
This theorem is referenced by:  gruxp  9219
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