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Theorem gruuni 8610
Description: A Grothendieck's universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.)
Assertion
Ref Expression
gruuni  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  U. A  e.  U )

Proof of Theorem gruuni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 uniiun 4087 . 2  |-  U. A  =  U_ x  e.  A  x
2 gruelss 8604 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
3 dfss3 3283 . . . 4  |-  ( A 
C_  U  <->  A. x  e.  A  x  e.  U )
42, 3sylib 189 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A. x  e.  A  x  e.  U )
5 gruiun 8609 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  x  e.  U )  ->  U_ x  e.  A  x  e.  U )
64, 5mpd3an3 1280 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  U_ x  e.  A  x  e.  U )
71, 6syl5eqel 2473 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  U. A  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2651    C_ wss 3265   U.cuni 3959   U_ciun 4037   Univcgru 8600
This theorem is referenced by:  gruwun  8623  gruina  8628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-gru 8601
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