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Theorem elpwuni 3782
Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
elpwuni  |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )

Proof of Theorem elpwuni
StepHypRef Expression
1 sspwuni 3780 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissel 3650 . . . 4  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
32expcom 114 . . 3  |-  ( B  e.  A  ->  ( U. A  C_  B  ->  U. A  =  B
) )
4 eqimss 3060 . . 3  |-  ( U. A  =  B  ->  U. A  C_  B )
53, 4impbid1 140 . 2  |-  ( B  e.  A  ->  ( U. A  C_  B  <->  U. A  =  B ) )
61, 5syl5bb 190 1  |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434    C_ wss 2982   ~Pcpw 3400   U.cuni 3621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-in 2988  df-ss 2995  df-pw 3402  df-uni 3622
This theorem is referenced by: (None)
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