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Theorem axpweq 3952
Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3955 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 3400 . . . 4  |-  ( ~P A  e.  _V  ->  ~P A  e.  ~P ~P A )
2 pweq 3390 . . . . . 6  |-  ( x  =  ~P A  ->  ~P x  =  ~P ~P A )
32eleq2d 2123 . . . . 5  |-  ( x  =  ~P A  -> 
( ~P A  e. 
~P x  <->  ~P A  e.  ~P ~P A ) )
43spcegv 2658 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  ~P ~P A  ->  E. x ~P A  e.  ~P x ) )
51, 4mpd 13 . . 3  |-  ( ~P A  e.  _V  ->  E. x ~P A  e. 
~P x )
6 elex 2583 . . . 4  |-  ( ~P A  e.  ~P x  ->  ~P A  e.  _V )
76exlimiv 1505 . . 3  |-  ( E. x ~P A  e. 
~P x  ->  ~P A  e.  _V )
85, 7impbii 121 . 2  |-  ( ~P A  e.  _V  <->  E. x ~P A  e.  ~P x )
9 vex 2577 . . . . 5  |-  x  e. 
_V
109elpw2 3939 . . . 4  |-  ( ~P A  e.  ~P x  <->  ~P A  C_  x )
11 pwss 3402 . . . . 5  |-  ( ~P A  C_  x  <->  A. y
( y  C_  A  ->  y  e.  x ) )
12 dfss2 2962 . . . . . . 7  |-  ( y 
C_  A  <->  A. z
( z  e.  y  ->  z  e.  A
) )
1312imbi1i 231 . . . . . 6  |-  ( ( y  C_  A  ->  y  e.  x )  <->  ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1413albii 1375 . . . . 5  |-  ( A. y ( y  C_  A  ->  y  e.  x
)  <->  A. y ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
1511, 14bitri 177 . . . 4  |-  ( ~P A  C_  x  <->  A. y
( A. z ( z  e.  y  -> 
z  e.  A )  ->  y  e.  x
) )
1610, 15bitri 177 . . 3  |-  ( ~P A  e.  ~P x  <->  A. y ( A. z
( z  e.  y  ->  z  e.  A
)  ->  y  e.  x ) )
1716exbii 1512 . 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 177 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 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389
This theorem is referenced by: (None)
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