Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpwgdedVD Unicode version

Theorem elpwgdedVD 28457
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 3721. In form of VD deduction with  ph and  ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 28077 is elpwgdedVD 28457 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpwgdedVD.1  |-  (. ph  ->.  A  e.  _V ).
elpwgdedVD.2  |-  (. ps  ->.  A 
C_  B ).
Assertion
Ref Expression
elpwgdedVD  |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).

Proof of Theorem elpwgdedVD
StepHypRef Expression
1 elpwgdedVD.1 . 2  |-  (. ph  ->.  A  e.  _V ).
2 elpwgdedVD.2 . 2  |-  (. ps  ->.  A 
C_  B ).
3 elpwg 3721 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimpar 471 . 2  |-  ( ( A  e.  _V  /\  A  C_  B )  ->  A  e.  ~P B
)
51, 2, 4el12 28263 1  |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).
Colors of variables: wff set class
Syntax hints:    e. wcel 1715   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   (.wvd1 28084   (.wvhc2 28096
This theorem is referenced by:  sspwimpVD  28459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-in 3245  df-ss 3252  df-pw 3716  df-vd1 28085  df-vhc2 28097
  Copyright terms: Public domain W3C validator