MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruun Structured version   Unicode version

Theorem gruun 8978
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 4228 . . 3  |-  U. { A ,  B }  =  U_ x  e.  { A ,  B }
x
2 uniprg 4110 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B )
)
323adant1 1006 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B ) )
41, 3syl5reqr 2490 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  =  U_ x  e.  { A ,  B }
x )
5 simp1 988 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
6 grupr 8969 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 vex 2980 . . . . . . 7  |-  x  e. 
_V
87elpr 3900 . . . . . 6  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
9 eleq1a 2512 . . . . . . 7  |-  ( A  e.  U  ->  (
x  =  A  ->  x  e.  U )
)
10 eleq1a 2512 . . . . . . 7  |-  ( B  e.  U  ->  (
x  =  B  ->  x  e.  U )
)
119, 10jaao 509 . . . . . 6  |-  ( ( A  e.  U  /\  B  e.  U )  ->  ( ( x  =  A  \/  x  =  B )  ->  x  e.  U ) )
128, 11syl5bi 217 . . . . 5  |-  ( ( A  e.  U  /\  B  e.  U )  ->  ( x  e.  { A ,  B }  ->  x  e.  U ) )
1312ralrimiv 2803 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B }
x  e.  U )
14133adant1 1006 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B } x  e.  U
)
15 gruiun 8971 . . 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 1218 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U_ x  e.  { A ,  B } x  e.  U
)
174, 16eqeltrd 2517 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 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720    u. cun 3331   {cpr 3884   U.cuni 4096   U_ciun 4176   Univcgru 8962
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-gru 8963
This theorem is referenced by:  gruxp  8979
  Copyright terms: Public domain W3C validator