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Theorem elpwpw 3903
Description: Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
elpwpw  |-  ( A  e.  ~P ~P B  <->  ( A  e.  _V  /\  U. A  C_  B )
)

Proof of Theorem elpwpw
StepHypRef Expression
1 elpwb 3521 . 2  |-  ( A  e.  ~P ~P B  <->  ( A  e.  _V  /\  A  C_  ~P B ) )
2 sspwuni 3901 . . 3  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
32anbi2i 453 . 2  |-  ( ( A  e.  _V  /\  A  C_  ~P B )  <-> 
( A  e.  _V  /\ 
U. A  C_  B
) )
41, 3bitri 183 1  |-  ( A  e.  ~P ~P B  <->  ( A  e.  _V  /\  U. A  C_  B )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1481   _Vcvv 2687    C_ wss 3072   ~Pcpw 3511   U.cuni 3740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2689  df-in 3078  df-ss 3085  df-pw 3513  df-uni 3741
This theorem is referenced by:  pwpwab  3904  elpwpwel  4400
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