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Theorem gruss 8418
Description: Any subset of an element of a Grothendieck's 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 4174 . . . 4  |-  ( A  e.  U  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 452 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 grupw 8417 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
4 gruelss 8416 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P A  e.  U )  ->  ~P A  C_  U
)
53, 4syldan 456 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  C_  U )
65sseld 3179 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  ->  B  e.  U ) )
72, 6sylbird 226 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  C_  A  ->  B  e.  U ) )
873impia 1148 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  C_  A )  ->  B  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   Univcgru 8412
This theorem is referenced by:  grurn  8423  gruima  8424  gruxp  8429  grumap  8430  gruixp  8431  gruiin  8432  grudomon  8439  gruina  8440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-tr 4114  df-iota 5219  df-fv 5263  df-ov 5861  df-gru 8413
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