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Theorem elpwgdedVD 27826
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 3606. In form of VD deduction with  ph and  ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 27466 is elpwgdedVD 27826 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 3606 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ~P B  <->  A 
C_  B ) )
43biimpar 473 . 2  |-  ( ( A  e.  _V  /\  A  C_  B )  ->  A  e.  ~P B
)
51, 2, 4el12 27634 1  |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   _Vcvv 2763    C_ wss 3127   ~Pcpw 3599   (.wvd1 27473   (.wvhc2 27485
This theorem is referenced by:  sspwimpVD  27828
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765  df-in 3134  df-ss 3141  df-pw 3601  df-vd1 27474  df-vhc2 27486
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