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Theorem gruss 9075
Description: Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.)
Assertion
Ref Expression
gruss  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  C_  A )  ->  B  e.  U )

Proof of Theorem gruss
StepHypRef Expression
1 elpw2g 4564 . . . 4  |-  ( A  e.  U  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 466 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 grupw 9074 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
4 gruelss 9073 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P A  e.  U )  ->  ~P A  C_  U
)
53, 4syldan 470 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  C_  U )
65sseld 3464 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  ->  B  e.  U ) )
72, 6sylbird 235 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  C_  A  ->  B  e.  U ) )
873impia 1185 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  C_  A )  ->  B  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    C_ wss 3437   ~Pcpw 3969   Univcgru 9069
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 1955  ax-ext 2432  ax-sep 4522
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 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-tr 4495  df-iota 5490  df-fv 5535  df-ov 6204  df-gru 9070
This theorem is referenced by:  grurn  9080  gruima  9081  gruxp  9086  grumap  9087  gruixp  9088  gruiin  9089  grudomon  9096  gruina  9097
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