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Theorem pwpwssunieq 3970
Description: The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
pwpwssunieq  |-  { x  |  U. x  =  A }  C_  ~P ~P A
Distinct variable group:    x, A

Proof of Theorem pwpwssunieq
StepHypRef Expression
1 eqimss 3207 . . 3  |-  ( U. x  =  A  ->  U. x  C_  A )
21ss2abi 3225 . 2  |-  { x  |  U. x  =  A }  C_  { x  |  U. x  C_  A }
3 pwpwab 3969 . 2  |-  ~P ~P A  =  { x  |  U. x  C_  A }
42, 3sseqtrri 3188 1  |-  { x  |  U. x  =  A }  C_  ~P ~P A
Colors of variables: wff set class
Syntax hints:    = wceq 1353   {cab 2161    C_ wss 3127   ~Pcpw 3572   U.cuni 3805
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-pw 3574  df-uni 3806
This theorem is referenced by:  toponsspwpwg  13013  dmtopon  13014
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