Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ump Unicode version

Theorem ump 25149
Description: The union of a part of a powerset belongs to it. (Contributed by FL, 16-Nov-2007.)
Assertion
Ref Expression
ump  |-  ( A  e.  V  ->  U. {
x  e.  ~P A  |  ph }  e.  ~P A )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem ump
StepHypRef Expression
1 ssrab2 3271 . . 3  |-  { x  e.  ~P A  |  ph }  C_  ~P A
2 sspwuni 4003 . . 3  |-  ( { x  e.  ~P A  |  ph }  C_  ~P A 
<-> 
U. { x  e. 
~P A  |  ph }  C_  A )
31, 2mpbi 199 . 2  |-  U. {
x  e.  ~P A  |  ph }  C_  A
4 elpw2g 4190 . 2  |-  ( A  e.  V  ->  ( U. { x  e.  ~P A  |  ph }  e.  ~P A  <->  U. { x  e. 
~P A  |  ph }  C_  A ) )
53, 4mpbiri 224 1  |-  ( A  e.  V  ->  U. {
x  e.  ~P A  |  ph }  e.  ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   {crab 2560    C_ wss 3165   ~Pcpw 3638   U.cuni 3843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844
  Copyright terms: Public domain W3C validator