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Theorem gruel 8512
Description: Any element of an element of a Grothendieck's 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 8503 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
21sseld 3255 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  A  ->  B  e.  U ) )
323impia 1148 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  A )  ->  B  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1710   Univcgru 8499
This theorem is referenced by:  gruf  8520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-tr 4193  df-iota 5298  df-fv 5342  df-ov 5945  df-gru 8500
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