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Theorem grutr 9203
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 9202 . . 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 243 . 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 976    e. wcel 1844   A.wral 2756   ~Pcpw 3957   {cpr 3976   U.cuni 4193   Tr wtr 4491   ran crn 4826  (class class class)co 6280    ^m cmap 7459   Univcgru 9200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-tr 4492  df-iota 5535  df-fv 5579  df-ov 6283  df-gru 9201
This theorem is referenced by:  gruelss  9204  gruwun  9223  intgru  9224  gruina  9228  grur1  9230  grutsk  9232
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