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Theorem pwnss 3935
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 2144 . . . . . . 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 389 . . . . . 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 625 . . . . 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 2341 . . . . . . 7  |-  ( x  e/  x  <->  -.  x  e.  x )
5 eleq12 2144 . . . . . . . . 9  |-  ( ( x  =  y  /\  x  =  y )  ->  ( x  e.  x  <->  y  e.  y ) )
65anidms 389 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
76notbid 625 . . . . . . 7  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
84, 7syl5bb 190 . . . . . 6  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
98cbvrabv 2601 . . . . 5  |-  { x  e.  A  |  x  e/  x }  =  {
y  e.  A  |  -.  y  e.  y }
103, 9elrab2 2752 . . . 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 1306 . . . 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 7 . . 3  |-  -.  {
x  e.  A  |  x  e/  x }  e.  A
13 ssel 2994 . . 3  |-  ( ~P A  C_  A  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  ->  { x  e.  A  |  x  e/  x }  e.  A )
)
1412, 13mtoi 623 . 2  |-  ( ~P A  C_  A  ->  -. 
{ x  e.  A  |  x  e/  x }  e.  ~P A
)
15 ssrab2 3080 . . 3  |-  { x  e.  A  |  x  e/  x }  C_  A
16 elpw2g 3933 . . 3  |-  ( A  e.  V  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  <->  { x  e.  A  |  x  e/  x }  C_  A ) )
1715, 16mpbiri 166 . 2  |-  ( A  e.  V  ->  { x  e.  A  |  x  e/  x }  e.  ~P A )
1814, 17nsyl3 589 1  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434    e/ wnel 2340   {crab 2353    C_ wss 2974   ~Pcpw 3384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-nel 2341  df-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-pw 3386
This theorem is referenced by:  pwne  3936  pwuninel2  5925
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