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Theorem elpwgdedVD 27383
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 3537. In form of VD deduction with  ph and  ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 27023 is elpwgdedVD 27383 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 3537 . . 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 27191 1  |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530   (.wvd1 27030   (.wvhc2 27042
This theorem is referenced by:  sspwimpVD  27385
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-pw 3532  df-vd1 27031  df-vhc2 27043
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