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Theorem gruuni 8667
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 4136 . 2  |-  U. A  =  U_ x  e.  A  x
2 gruelss 8661 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
3 dfss3 3330 . . . 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 8666 . . 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 2519 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 1725   A.wral 2697    C_ wss 3312   U.cuni 4007   U_ciun 4085   Univcgru 8657
This theorem is referenced by:  gruwun  8680  gruina  8685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-gru 8658
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