ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elpw2g Unicode version

Theorem elpw2g 4051
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 3489 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
2 ssexg 4037 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
3 elpwg 3488 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimparc 297 . . . 4  |-  ( ( A  C_  B  /\  A  e.  _V )  ->  A  e.  ~P B
)
52, 4syldan 280 . . 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 1465   _Vcvv 2660    C_ wss 3041   ~Pcpw 3480
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by:  elpw2  4052  pwnss  4053  elfir  6829  istopg  12093  uniopn  12095  iscld  12199  ntrval  12206  clsval  12207  discld  12232  neival  12239  isnei  12240  restdis  12280  cnpfval  12291  cndis  12337  blfvalps  12481  blfps  12505  blf  12506  reldvg  12744
  Copyright terms: Public domain W3C validator