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Theorem grutr 9167
Description: A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
grutr  |-  ( U  e.  Univ  ->  Tr  U
)

Proof of Theorem grutr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elgrug 9166 . . 3  |-  ( U  e.  Univ  ->  ( U  e.  Univ  <->  ( Tr  U  /\  A. x  e.  U  ( ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U  /\  A. y  e.  ( U  ^m  x ) U. ran  y  e.  U
) ) ) )
21ibi 241 . 2  |-  ( U  e.  Univ  ->  ( Tr  U  /\  A. x  e.  U  ( ~P x  e.  U  /\  A. y  e.  U  {
x ,  y }  e.  U  /\  A. y  e.  ( U  ^m  x ) U. ran  y  e.  U )
) )
32simpld 459 1  |-  ( U  e.  Univ  ->  Tr  U
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814   ~Pcpw 4010   {cpr 4029   U.cuni 4245   Tr wtr 4540   ran crn 5000  (class class class)co 6282    ^m cmap 7417   Univcgru 9164
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-tr 4541  df-iota 5549  df-fv 5594  df-ov 6285  df-gru 9165
This theorem is referenced by:  gruelss  9168  gruwun  9187  intgru  9188  gruina  9192  grur1  9194  grutsk  9196
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