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Theorem gruelss 8416
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 8415 . 2  |-  ( U  e.  Univ  ->  Tr  U
)
2 trss 4122 . . 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 1684    C_ wss 3152   Tr wtr 4113   Univcgru 8412
This theorem is referenced by:  gruss  8418  gruuni  8422  gruel  8425  gruina  8440  grur1a  8441  grur1  8442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-tr 4114  df-iota 5219  df-fv 5263  df-ov 5861  df-gru 8413
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