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Theorem gruel 8966
Description: Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruel  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  A )  ->  B  e.  U )

Proof of Theorem gruel
StepHypRef Expression
1 gruelss 8957 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
21sseld 3352 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  A  ->  B  e.  U ) )
323impia 1179 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  A )  ->  B  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    e. wcel 1761   Univcgru 8953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-tr 4383  df-iota 5378  df-fv 5423  df-ov 6093  df-gru 8954
This theorem is referenced by:  gruf  8974
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