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Theorem gruelss 8432
Description: A Grothendieck's universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruelss  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )

Proof of Theorem gruelss
StepHypRef Expression
1 grutr 8431 . 2  |-  ( U  e.  Univ  ->  Tr  U
)
2 trss 4138 . . 3  |-  ( Tr  U  ->  ( A  e.  U  ->  A  C_  U ) )
32imp 418 . 2  |-  ( ( Tr  U  /\  A  e.  U )  ->  A  C_  U )
41, 3sylan 457 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    C_ wss 3165   Tr wtr 4129   Univcgru 8428
This theorem is referenced by:  gruss  8434  gruuni  8438  gruel  8441  gruina  8456  grur1a  8457  grur1  8458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-tr 4130  df-iota 5235  df-fv 5279  df-ov 5877  df-gru 8429
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