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Theorem elpwuni 3989
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 3987 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissel 3856 . . . 4  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
32expcom 424 . . 3  |-  ( B  e.  A  ->  ( U. A  C_  B  ->  U. A  =  B
) )
4 eqimss 3230 . . 3  |-  ( U. A  =  B  ->  U. A  C_  B )
53, 4impbid1 194 . 2  |-  ( B  e.  A  ->  ( U. A  C_  B  <->  U. A  =  B ) )
61, 5syl5bb 248 1  |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  mreuni  13502  issgon  23484  br2base  23574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
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