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Theorem gruun 8444
Description: A Grothendieck's 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 3971 . . 3  |-  U. { A ,  B }  =  U_ x  e.  { A ,  B }
x
2 uniprg 3858 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B )
)
323adant1 973 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U. { A ,  B }  =  ( A  u.  B ) )
41, 3syl5reqr 2343 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  =  U_ x  e.  { A ,  B }
x )
5 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U  e.  Univ )
6 grupr 8435 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  { A ,  B }  e.  U
)
7 vex 2804 . . . . . . 7  |-  x  e. 
_V
87elpr 3671 . . . . . 6  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
9 eleq1a 2365 . . . . . . 7  |-  ( A  e.  U  ->  (
x  =  A  ->  x  e.  U )
)
10 eleq1a 2365 . . . . . . 7  |-  ( B  e.  U  ->  (
x  =  B  ->  x  e.  U )
)
119, 10jaao 495 . . . . . 6  |-  ( ( A  e.  U  /\  B  e.  U )  ->  ( ( x  =  A  \/  x  =  B )  ->  x  e.  U ) )
128, 11syl5bi 208 . . . . 5  |-  ( ( A  e.  U  /\  B  e.  U )  ->  ( x  e.  { A ,  B }  ->  x  e.  U ) )
1312ralrimiv 2638 . . . 4  |-  ( ( A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B }
x  e.  U )
14133adant1 973 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  A. x  e.  { A ,  B } x  e.  U
)
15 gruiun 8437 . . 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 1182 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  U_ x  e.  { A ,  B } x  e.  U
)
174, 16eqeltrd 2370 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  U )  ->  ( A  u.  B )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163   {cpr 3654   U.cuni 3843   U_ciun 3921   Univcgru 8428
This theorem is referenced by:  gruxp  8445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-gru 8429
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