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Theorem elpwgded 28629
Description: elpwgdedVD 29009 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpwgded.1  |-  ( ph  ->  A  e.  _V )
elpwgded.2  |-  ( ps 
->  A  C_  B )
Assertion
Ref Expression
elpwgded  |-  ( (
ph  /\  ps )  ->  A  e.  ~P B
)

Proof of Theorem elpwgded
StepHypRef Expression
1 elpwgded.1 . 2  |-  ( ph  ->  A  e.  _V )
2 elpwgded.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, 4syl2an 463 1  |-  ( (
ph  /\  ps )  ->  A  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638
This theorem is referenced by:  sspwimp  29010  sspwimpALT  29017
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
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