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Theorem elpwgded 28725
Description: elpwgdedVD 29103 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 3808 . . 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, 4syl2an 465 1  |-  ( (
ph  /\  ps )  ->  A  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801
This theorem is referenced by:  sspwimp  29104  sspwimpALT  29111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803
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