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Theorem ump 25046
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 3258 . . 3  |-  { x  e.  ~P A  |  ph }  C_  ~P A
2 sspwuni 3987 . . 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 4174 . 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 1684   {crab 2547    C_ wss 3152   ~Pcpw 3625   U.cuni 3827
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-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-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
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