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Theorem grutr 8603
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 8602 . . 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 1717   A.wral 2651   ~Pcpw 3744   {cpr 3760   U.cuni 3959   Tr wtr 4245   ran crn 4821  (class class class)co 6022    ^m cmap 6956   Univcgru 8600
This theorem is referenced by:  gruelss  8604  gruwun  8623  intgru  8624  gruina  8628  grur1  8630  grutsk  8632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-tr 4246  df-iota 5360  df-fv 5404  df-ov 6025  df-gru 8601
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