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Theorem axpweq 4614
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4615 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 4012 . . . 4  |-  ( ~P A  e.  _V  ->  ~P A  e.  ~P ~P A )
2 pweq 4002 . . . . . 6  |-  ( x  =  ~P A  ->  ~P x  =  ~P ~P A )
32eleq2d 2524 . . . . 5  |-  ( x  =  ~P A  -> 
( ~P A  e. 
~P x  <->  ~P A  e.  ~P ~P A ) )
43spcegv 3192 . . . 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 3115 . . . 4  |-  ( ~P A  e.  ~P x  ->  ~P A  e.  _V )
76exlimiv 1727 . . 3  |-  ( E. x ~P A  e. 
~P x  ->  ~P A  e.  _V )
85, 7impbii 188 . 2  |-  ( ~P A  e.  _V  <->  E. x ~P A  e.  ~P x )
9 vex 3109 . . . . 5  |-  x  e. 
_V
109elpw2 4601 . . . 4  |-  ( ~P A  e.  ~P x  <->  ~P A  C_  x )
11 pwss 4014 . . . . 5  |-  ( ~P A  C_  x  <->  A. y
( y  C_  A  ->  y  e.  x ) )
12 dfss2 3478 . . . . . . 7  |-  ( y 
C_  A  <->  A. z
( z  e.  y  ->  z  e.  A
) )
1312imbi1i 323 . . . . . 6  |-  ( ( y  C_  A  ->  y  e.  x )  <->  ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1413albii 1645 . . . . 5  |-  ( A. y ( y  C_  A  ->  y  e.  x
)  <->  A. y ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1511, 14bitri 249 . . . 4  |-  ( ~P A  C_  x  <->  A. y
( A. z ( z  e.  y  -> 
z  e.  A )  ->  y  e.  x
) )
1610, 15bitri 249 . . 3  |-  ( ~P A  e.  ~P x  <->  A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
1716exbii 1672 . 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 249 1  |-  ( ~P A  e.  _V  <->  E. x A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001
This theorem is referenced by: (None)
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