MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gruss Structured version   Unicode version

Theorem gruss 8671
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 4363 . . . 4  |-  ( A  e.  U  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 453 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 grupw 8670 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  e.  U )
4 gruelss 8669 . . . . 5  |-  ( ( U  e.  Univ  /\  ~P A  e.  U )  ->  ~P A  C_  U
)
53, 4syldan 457 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ~P A  C_  U )
65sseld 3347 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  e.  ~P A  ->  B  e.  U ) )
72, 6sylbird 227 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( B  C_  A  ->  B  e.  U ) )
873impia 1150 1  |-  ( ( U  e.  Univ  /\  A  e.  U  /\  B  C_  A )  ->  B  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725    C_ wss 3320   ~Pcpw 3799   Univcgru 8665
This theorem is referenced by:  grurn  8676  gruima  8677  gruxp  8682  grumap  8683  gruixp  8684  gruiin  8685  grudomon  8692  gruina  8693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-tr 4303  df-iota 5418  df-fv 5462  df-ov 6084  df-gru 8666
  Copyright terms: Public domain W3C validator