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Theorem axpweq 4378
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4379 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
Hypothesis
Ref Expression
axpweq.1  |-  A  e. 
_V
Assertion
Ref Expression
axpweq  |-  ( ~P A  e.  _V  <->  E. x A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
Distinct variable group:    x, y, z, A

Proof of Theorem axpweq
StepHypRef Expression
1 pwidg 3813 . . . 4  |-  ( ~P A  e.  _V  ->  ~P A  e.  ~P ~P A )
2 pweq 3804 . . . . . 6  |-  ( x  =  ~P A  ->  ~P x  =  ~P ~P A )
32eleq2d 2505 . . . . 5  |-  ( x  =  ~P A  -> 
( ~P A  e. 
~P x  <->  ~P A  e.  ~P ~P A ) )
43spcegv 3039 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  ~P ~P A  ->  E. x ~P A  e.  ~P x ) )
51, 4mpd 15 . . 3  |-  ( ~P A  e.  _V  ->  E. x ~P A  e. 
~P x )
6 elex 2966 . . . 4  |-  ( ~P A  e.  ~P x  ->  ~P A  e.  _V )
76exlimiv 1645 . . 3  |-  ( E. x ~P A  e. 
~P x  ->  ~P A  e.  _V )
85, 7impbii 182 . 2  |-  ( ~P A  e.  _V  <->  E. x ~P A  e.  ~P x )
9 vex 2961 . . . . 5  |-  x  e. 
_V
109elpw2 4366 . . . 4  |-  ( ~P A  e.  ~P x  <->  ~P A  C_  x )
11 pwss 3815 . . . . 5  |-  ( ~P A  C_  x  <->  A. y
( y  C_  A  ->  y  e.  x ) )
12 dfss2 3339 . . . . . . 7  |-  ( y 
C_  A  <->  A. z
( z  e.  y  ->  z  e.  A
) )
1312imbi1i 317 . . . . . 6  |-  ( ( y  C_  A  ->  y  e.  x )  <->  ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1413albii 1576 . . . . 5  |-  ( A. y ( y  C_  A  ->  y  e.  x
)  <->  A. y ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1511, 14bitri 242 . . . 4  |-  ( ~P A  C_  x  <->  A. y
( A. z ( z  e.  y  -> 
z  e.  A )  ->  y  e.  x
) )
1610, 15bitri 242 . . 3  |-  ( ~P A  e.  ~P x  <->  A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
1716exbii 1593 . 2  |-  ( E. x ~P A  e. 
~P x  <->  E. x A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
188, 17bitri 242 1  |-  ( ~P A  e.  _V  <->  E. x A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-sep 4332
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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