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Theorem gruuni 9190
Description: A Grothendieck 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 4384 . 2  |-  U. A  =  U_ x  e.  A  x
2 gruelss 9184 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
3 dfss3 3499 . . . 4  |-  ( A 
C_  U  <->  A. x  e.  A  x  e.  U )
42, 3sylib 196 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A. x  e.  A  x  e.  U )
5 gruiun 9189 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  x  e.  U )  ->  U_ x  e.  A  x  e.  U )
64, 5mpd3an3 1325 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  U_ x  e.  A  x  e.  U )
71, 6syl5eqel 2559 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  U. A  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2817    C_ wss 3481   U.cuni 4251   U_ciun 4331   Univcgru 9180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-gru 9181
This theorem is referenced by:  gruwun  9203  gruina  9208
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