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Theorem gruelss 8669
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 8668 . 2  |-  ( U  e.  Univ  ->  Tr  U
)
2 trss 4311 . . 3  |-  ( Tr  U  ->  ( A  e.  U  ->  A  C_  U ) )
32imp 419 . 2  |-  ( ( Tr  U  /\  A  e.  U )  ->  A  C_  U )
41, 3sylan 458 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    C_ wss 3320   Tr wtr 4302   Univcgru 8665
This theorem is referenced by:  gruss  8671  gruuni  8675  gruel  8678  grur1a  8694  grur1  8695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-tr 4303  df-iota 5418  df-fv 5462  df-ov 6084  df-gru 8666
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