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Theorem gruelss 9237
Description: A Grothendieck 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 9236 . 2  |-  ( U  e.  Univ  ->  Tr  U
)
2 trss 4499 . . 3  |-  ( Tr  U  ->  ( A  e.  U  ->  A  C_  U ) )
32imp 436 . 2  |-  ( ( Tr  U  /\  A  e.  U )  ->  A  C_  U )
41, 3sylan 479 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904    C_ wss 3390   Tr wtr 4490   Univcgru 9233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-tr 4491  df-iota 5553  df-fv 5597  df-ov 6311  df-gru 9234
This theorem is referenced by:  gruss  9239  gruuni  9243  gruel  9246  grur1a  9262  grur1  9263
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