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Theorem axpweq 4187
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4188 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 3637 . . . 4  |-  ( ~P A  e.  _V  ->  ~P A  e.  ~P ~P A )
2 pweq 3628 . . . . . 6  |-  ( x  =  ~P A  ->  ~P x  =  ~P ~P A )
32eleq2d 2350 . . . . 5  |-  ( x  =  ~P A  -> 
( ~P A  e. 
~P x  <->  ~P A  e.  ~P ~P A ) )
43spcegv 2869 . . . 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 2796 . . . 4  |-  ( ~P A  e.  ~P x  ->  ~P A  e.  _V )
76exlimiv 1666 . . 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 2791 . . . . 5  |-  x  e. 
_V
109elpw2 4175 . . . 4  |-  ( ~P A  e.  ~P x  <->  ~P A  C_  x )
11 pwss 3639 . . . . 5  |-  ( ~P A  C_  x  <->  A. y
( y  C_  A  ->  y  e.  x ) )
12 dfss2 3169 . . . . . . 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 1553 . . . . 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 1569 . 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 1527   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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