Theorem elpwgdedVD 39153

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4166. In form of VD deduction with ph and ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 38780 is elpwgdedVD 39153 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

Step	Hyp	Ref	Expression
1		elpwgdedVD.1	. 2 |- (. ph ->. A e. _V ).
2		elpwgdedVD.2	. 2 |- (. ps ->. A C_ B ).
3		elpwg 4166	. . 3 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
4	3	biimpar 502	. 2 |- ( ( A e. _V /\ A C_ B ) -> A e. ~P B )
5	1, 2, 4	el12 38953	1 |- (. (. ph ,. ps ). ->. A e. ~P B ).

Syntax hints:   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   (.wvd1 38785   (.wvhc2 38796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-vd1 38786  df-vhc2 38797

This theorem is referenced by:  sspwimpVD  39155