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Theorem gruss 9107
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 4545 . . . 4  |-  ( A  e.  U  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 464 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 grupw 9106 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
4 gruelss 9105 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P A  e.  U )  ->  ~P A  C_  U
)
53, 4syldan 468 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  C_  U )
65sseld 3433 . . 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 1191 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 367    /\ w3a 971    e. wcel 1836    C_ wss 3406   ~Pcpw 3944   Univcgru 9101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-tr 4478  df-iota 5477  df-fv 5521  df-ov 6221  df-gru 9102
This theorem is referenced by:  grurn  9112  gruima  9113  gruxp  9118  grumap  9119  gruixp  9120  gruiin  9121  grudomon  9128  gruina  9129
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