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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

Proof of Theorem elpwpw
StepHypRef Expression
1 elpwb 3521 . 2
2 sspwuni 3901 . . 3
32anbi2i 453 . 2
41, 3bitri 183 1
 Colors of variables: wff set class Syntax hints:   wa 103   wb 104   wcel 1481  cvv 2687   wss 3072  cpw 3511  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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