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

Theorem gruun 9173
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 4368 . . 3  |-  U. { A ,  B }  =  U_ x  e.  { A ,  B }
x
2 uniprg 4249 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B )
)
323adant1 1012 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B ) )
41, 3syl5reqr 2510 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  =  U_ x  e.  { A ,  B }
x )
5 simp1 994 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
6 grupr 9164 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 vex 3109 . . . . . . 7  |-  x  e. 
_V
87elpr 4034 . . . . . 6  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
9 eleq1a 2537 . . . . . . 7  |-  ( A  e.  U  ->  (
x  =  A  ->  x  e.  U )
)
10 eleq1a 2537 . . . . . . 7  |-  ( B  e.  U  ->  (
x  =  B  ->  x  e.  U )
)
119, 10jaao 507 . . . . . 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 2866 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B }
x  e.  U )
14133adant1 1012 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B } x  e.  U
)
15 gruiun 9166 . . 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 1226 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U_ x  e.  { A ,  B } x  e.  U
)
174, 16eqeltrd 2542 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    u. cun 3459   {cpr 4018   U.cuni 4235   U_ciun 4315   Univcgru 9157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-gru 9158
This theorem is referenced by:  gruxp  9174
  Copyright terms: Public domain W3C validator