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

Theorem gruuni 8963
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 4220 . 2  |-  U. A  =  U_ x  e.  A  x
2 gruelss 8957 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
3 dfss3 3343 . . . 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 8962 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  A. x  e.  A  x  e.  U )  ->  U_ x  e.  A  x  e.  U )
64, 5mpd3an3 1310 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  U_ x  e.  A  x  e.  U )
71, 6syl5eqel 2525 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 1761   A.wral 2713    C_ wss 3325   U.cuni 4088   U_ciun 4168   Univcgru 8953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-gru 8954
This theorem is referenced by:  gruwun  8976  gruina  8981
  Copyright terms: Public domain W3C validator