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Theorem axpweq 4203
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4204 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 3650 . . . 4  |-  ( ~P A  e.  _V  ->  ~P A  e.  ~P ~P A )
2 pweq 3641 . . . . . 6  |-  ( x  =  ~P A  ->  ~P x  =  ~P ~P A )
32eleq2d 2363 . . . . 5  |-  ( x  =  ~P A  -> 
( ~P A  e. 
~P x  <->  ~P A  e.  ~P ~P A ) )
43spcegv 2882 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  ~P ~P A  ->  E. x ~P A  e.  ~P x ) )
51, 4mpd 14 . . 3  |-  ( ~P A  e.  _V  ->  E. x ~P A  e. 
~P x )
6 elex 2809 . . . 4  |-  ( ~P A  e.  ~P x  ->  ~P A  e.  _V )
76exlimiv 1624 . . 3  |-  ( E. x ~P A  e. 
~P x  ->  ~P A  e.  _V )
85, 7impbii 180 . 2  |-  ( ~P A  e.  _V  <->  E. x ~P A  e.  ~P x )
9 vex 2804 . . . . 5  |-  x  e. 
_V
109elpw2 4191 . . . 4  |-  ( ~P A  e.  ~P x  <->  ~P A  C_  x )
11 pwss 3652 . . . . 5  |-  ( ~P A  C_  x  <->  A. y
( y  C_  A  ->  y  e.  x ) )
12 dfss2 3182 . . . . . . 7  |-  ( y 
C_  A  <->  A. z
( z  e.  y  ->  z  e.  A
) )
1312imbi1i 315 . . . . . 6  |-  ( ( y  C_  A  ->  y  e.  x )  <->  ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1413albii 1556 . . . . 5  |-  ( A. y ( y  C_  A  ->  y  e.  x
)  <->  A. y ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1511, 14bitri 240 . . . 4  |-  ( ~P A  C_  x  <->  A. y
( A. z ( z  e.  y  -> 
z  e.  A )  ->  y  e.  x
) )
1610, 15bitri 240 . . 3  |-  ( ~P A  e.  ~P x  <->  A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
1716exbii 1572 . 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 240 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 176   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638
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  ax-sep 4157
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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