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Theorem gruelss 9064
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 9063 . 2  |-  ( U  e.  Univ  ->  Tr  U
)
2 trss 4494 . . 3  |-  ( Tr  U  ->  ( A  e.  U  ->  A  C_  U ) )
32imp 429 . 2  |-  ( ( Tr  U  /\  A  e.  U )  ->  A  C_  U )
41, 3sylan 471 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    C_ wss 3428   Tr wtr 4485   Univcgru 9060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-tr 4486  df-iota 5481  df-fv 5526  df-ov 6195  df-gru 9061
This theorem is referenced by:  gruss  9066  gruuni  9070  gruel  9073  grur1a  9089  grur1  9090
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