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Theorem elpwuni 3962
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 3957 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissel 3825 . . . 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 3201 . . 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 1348    e. wcel 2141    C_ wss 3121   ~Pcpw 3566   U.cuni 3796
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797
This theorem is referenced by: (None)
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