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Theorem elpwgdedVD 29009
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 3645. In form of VD deduction with  ph and  ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 28629 is elpwgdedVD 29009 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 3645 . . 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 28815 1  |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   (.wvd1 28636   (.wvhc2 28648
This theorem is referenced by:  sspwimpVD  29011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-vd1 28637  df-vhc2 28649
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