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Theorem elpw2g 4049
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
Assertion
Ref Expression
elpw2g  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )

Proof of Theorem elpw2g
StepHypRef Expression
1 elpwi 3487 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
2 ssexg 4035 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
3 elpwg 3486 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimparc 295 . . . 4  |-  ( ( A  C_  B  /\  A  e.  _V )  ->  A  e.  ~P B
)
52, 4syldan 278 . . 3  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  ~P B
)
65expcom 115 . 2  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  ~P B ) )
71, 6impbid2 142 1  |-  ( B  e.  V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1463   _Vcvv 2658    C_ wss 3039   ~Pcpw 3478
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480
This theorem is referenced by:  elpw2  4050  pwnss  4051  elfir  6827  istopg  12061  uniopn  12063  iscld  12167  ntrval  12174  clsval  12175  discld  12200  neival  12207  isnei  12208  restdis  12248  cnpfval  12259  cndis  12305  blfvalps  12449  blfps  12473  blf  12474  reldvg  12691
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