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Theorem gruel 8638
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 8629 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  C_  U )
21sseld 3311 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  A  ->  B  e.  U ) )
323impia 1150 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  e.  A )  ->  B  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   Univcgru 8625
This theorem is referenced by:  gruf  8646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-tr 4267  df-iota 5381  df-fv 5425  df-ov 6047  df-gru 8626
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