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Theorem elpwpw 3970
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 3584 . 2  |-  ( A  e.  ~P ~P B  <->  ( A  e.  _V  /\  A  C_  ~P B ) )
2 sspwuni 3968 . . 3  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
32anbi2i 457 . 2  |-  ( ( A  e.  _V  /\  A  C_  ~P B )  <-> 
( A  e.  _V  /\ 
U. A  C_  B
) )
41, 3bitri 184 1  |-  ( A  e.  ~P ~P B  <->  ( A  e.  _V  /\  U. A  C_  B )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2148   _Vcvv 2737    C_ wss 3129   ~Pcpw 3574   U.cuni 3807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-uni 3808
This theorem is referenced by:  pwpwab  3971  elpwpwel  4471
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