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Theorem grutr 8658
Description: A Grothendieck's 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 8657 . . 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 233 . 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 446 1  |-  ( U  e.  Univ  ->  Tr  U
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2697   ~Pcpw 3791   {cpr 3807   U.cuni 4007   Tr wtr 4294   ran crn 4871  (class class class)co 6073    ^m cmap 7010   Univcgru 8655
This theorem is referenced by:  gruelss  8659  gruwun  8678  intgru  8679  gruina  8683  grur1  8685  grutsk  8687
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-tr 4295  df-iota 5410  df-fv 5454  df-ov 6076  df-gru 8656
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