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Theorem grothpwex 9097
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 9093. Note that ax-pow 4570 is not used by the proof. Use axpweq 4569 to obtain ax-pow 4570. Use pwex 4575 or pwexg 4576 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.)
Assertion
Ref Expression
grothpwex  |-  ~P x  e.  _V

Proof of Theorem grothpwex
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 9094 . 2  |-  E. y
( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )
2 simpl 457 . . . . . . . 8  |-  ( ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  ->  ~P z  C_  y )
32ralimi 2811 . . . . . . 7  |-  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  ->  A. z  e.  y  ~P z  C_  y
)
4 pweq 3963 . . . . . . . . 9  |-  ( z  =  x  ->  ~P z  =  ~P x
)
54sseq1d 3483 . . . . . . . 8  |-  ( z  =  x  ->  ( ~P z  C_  y  <->  ~P x  C_  y ) )
65rspccv 3168 . . . . . . 7  |-  ( A. z  e.  y  ~P z  C_  y  ->  (
x  e.  y  ->  ~P x  C_  y ) )
73, 6syl 16 . . . . . 6  |-  ( A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  ->  ( x  e.  y  ->  ~P x  C_  y ) )
87anim2i 569 . . . . 5  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
) )  ->  (
x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y
) ) )
983adant3 1008 . . . 4  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  /\  A. z  e.  ~P  y ( z 
~~  y  \/  z  e.  y ) )  -> 
( x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y ) ) )
10 pm3.35 587 . . . 4  |-  ( ( x  e.  y  /\  ( x  e.  y  ->  ~P x  C_  y
) )  ->  ~P x  C_  y )
11 vex 3073 . . . . 5  |-  y  e. 
_V
1211ssex 4536 . . . 4  |-  ( ~P x  C_  y  ->  ~P x  e.  _V )
139, 10, 123syl 20 . . 3  |-  ( ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w
)  /\  A. z  e.  ~P  y ( z 
~~  y  \/  z  e.  y ) )  ->  ~P x  e.  _V )
1413exlimiv 1689 . 2  |-  ( E. y ( x  e.  y  /\  A. z  e.  y  ( ~P z  C_  y  /\  E. w  e.  y  ~P z  C_  w )  /\  A. z  e.  ~P  y
( z  ~~  y  \/  z  e.  y
) )  ->  ~P x  e.  _V )
151, 14ax-mp 5 1  |-  ~P x  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965   E.wex 1587    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070    C_ wss 3428   ~Pcpw 3960   class class class wbr 4392    ~~ cen 7409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-groth 9093
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3072  df-in 3435  df-ss 3442  df-pw 3962
This theorem is referenced by:  isrnsigaOLD  26691
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