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Theorem elpwgdedVD 29029
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 3806. In form of VD deduction with  ph and  ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 28651 is elpwgdedVD 29029 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 3806 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimpar 472 . 2  |-  ( ( A  e.  _V  /\  A  C_  B )  ->  A  e.  ~P B
)
51, 2, 4el12 28838 1  |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   (.wvd1 28660   (.wvhc2 28672
This theorem is referenced by:  sspwimpVD  29031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-vd1 28661  df-vhc2 28673
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