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Theorem pwnss 4176
Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwnss  |-  ( A  e.  V  ->  -.  ~P A  C_  A )

Proof of Theorem pwnss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq12 2345 . . . . . . 7  |-  ( ( y  =  { x  e.  A  |  x  e/  x }  /\  y  =  { x  e.  A  |  x  e/  x } )  ->  (
y  e.  y  <->  { x  e.  A  |  x  e/  x }  e.  {
x  e.  A  |  x  e/  x } ) )
21anidms 626 . . . . . 6  |-  ( y  =  { x  e.  A  |  x  e/  x }  ->  ( y  e.  y  <->  { x  e.  A  |  x  e/  x }  e.  {
x  e.  A  |  x  e/  x } ) )
32notbid 285 . . . . 5  |-  ( y  =  { x  e.  A  |  x  e/  x }  ->  ( -.  y  e.  y  <->  -.  { x  e.  A  |  x  e/  x }  e.  {
x  e.  A  |  x  e/  x } ) )
4 df-nel 2449 . . . . . . 7  |-  ( x  e/  x  <->  -.  x  e.  x )
5 eleq12 2345 . . . . . . . . 9  |-  ( ( x  =  y  /\  x  =  y )  ->  ( x  e.  x  <->  y  e.  y ) )
65anidms 626 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
76notbid 285 . . . . . . 7  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
84, 7syl5bb 248 . . . . . 6  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
98cbvrabv 2787 . . . . 5  |-  { x  e.  A  |  x  e/  x }  =  {
y  e.  A  |  -.  y  e.  y }
103, 9elrab2 2925 . . . 4  |-  ( { x  e.  A  |  x  e/  x }  e.  { x  e.  A  |  x  e/  x }  <->  ( {
x  e.  A  |  x  e/  x }  e.  A  /\  -.  { x  e.  A  |  x  e/  x }  e.  {
x  e.  A  |  x  e/  x } ) )
11 pclem6 896 . . . 4  |-  ( ( { x  e.  A  |  x  e/  x }  e.  { x  e.  A  |  x  e/  x }  <->  ( {
x  e.  A  |  x  e/  x }  e.  A  /\  -.  { x  e.  A  |  x  e/  x }  e.  {
x  e.  A  |  x  e/  x } ) )  ->  -.  { x  e.  A  |  x  e/  x }  e.  A
)
1210, 11ax-mp 8 . . 3  |-  -.  {
x  e.  A  |  x  e/  x }  e.  A
13 ssel 3174 . . 3  |-  ( ~P A  C_  A  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  ->  { x  e.  A  |  x  e/  x }  e.  A )
)
1412, 13mtoi 169 . 2  |-  ( ~P A  C_  A  ->  -. 
{ x  e.  A  |  x  e/  x }  e.  ~P A
)
15 ssrab2 3258 . . 3  |-  { x  e.  A  |  x  e/  x }  C_  A
16 elpw2g 4174 . . 3  |-  ( A  e.  V  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  <->  { x  e.  A  |  x  e/  x }  C_  A ) )
1715, 16mpbiri 224 . 2  |-  ( A  e.  V  ->  { x  e.  A  |  x  e/  x }  e.  ~P A )
1814, 17nsyl3 111 1  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    e/ wnel 2447   {crab 2547    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  pwne  4177  pwuninel2  6299
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-nel 2449  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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