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Theorem elpwgded 28370
Description: elpwgdedVD 28747 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 3774 . . 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, 4syl2an 464 1  |-  ( (
ph  /\  ps )  ->  A  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767
This theorem is referenced by:  sspwimp  28748  sspwimpALT  28755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-in 3295  df-ss 3302  df-pw 3769
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