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Theorem elpwuni 4180
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 4178 . 2  |-  ( A 
C_  ~P B  <->  U. A  C_  B )
2 unissel 4046 . . . 4  |-  ( ( U. A  C_  B  /\  B  e.  A
)  ->  U. A  =  B )
32expcom 426 . . 3  |-  ( B  e.  A  ->  ( U. A  C_  B  ->  U. A  =  B
) )
4 eqimss 3402 . . 3  |-  ( U. A  =  B  ->  U. A  C_  B )
53, 4impbid1 196 . 2  |-  ( B  e.  A  ->  ( U. A  C_  B  <->  U. A  =  B ) )
61, 5syl5bb 250 1  |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   U.cuni 4017
This theorem is referenced by:  mreuni  13827  ustuni  18258  utopbas  18267  issgon  24508  br2base  24621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-uni 4018
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