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Theorem elpwuni 3866
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 3861 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissel 3729 . . . 4  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
32expcom 115 . . 3  |-  ( B  e.  A  ->  ( U. A  C_  B  ->  U. A  =  B
) )
4 eqimss 3115 . . 3  |-  ( U. A  =  B  ->  U. A  C_  B )
53, 4impbid1 141 . 2  |-  ( B  e.  A  ->  ( U. A  C_  B  <->  U. A  =  B ) )
61, 5syl5bb 191 1  |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1312    e. wcel 1461    C_ wss 3035   ~Pcpw 3474   U.cuni 3700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476  df-uni 3701
This theorem is referenced by: (None)
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