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Theorem pwnss 4086
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 2204 . . . . . . 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 394 . . . . . 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 656 . . . . 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 2404 . . . . . . 7  |-  ( x  e/  x  <->  -.  x  e.  x )
5 eleq12 2204 . . . . . . . . 9  |-  ( ( x  =  y  /\  x  =  y )  ->  ( x  e.  x  <->  y  e.  y ) )
65anidms 394 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
76notbid 656 . . . . . . 7  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
84, 7syl5bb 191 . . . . . 6  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
98cbvrabv 2685 . . . . 5  |-  { x  e.  A  |  x  e/  x }  =  {
y  e.  A  |  -.  y  e.  y }
103, 9elrab2 2843 . . . 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 1352 . . . 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 5 . . 3  |-  -.  {
x  e.  A  |  x  e/  x }  e.  A
13 ssel 3091 . . 3  |-  ( ~P A  C_  A  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  ->  { x  e.  A  |  x  e/  x }  e.  A )
)
1412, 13mtoi 653 . 2  |-  ( ~P A  C_  A  ->  -. 
{ x  e.  A  |  x  e/  x }  e.  ~P A
)
15 ssrab2 3182 . . 3  |-  { x  e.  A  |  x  e/  x }  C_  A
16 elpw2g 4084 . . 3  |-  ( A  e.  V  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  <->  { x  e.  A  |  x  e/  x }  C_  A ) )
1715, 16mpbiri 167 . 2  |-  ( A  e.  V  ->  { x  e.  A  |  x  e/  x }  e.  ~P A )
1814, 17nsyl3 615 1  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480    e/ wnel 2403   {crab 2420    C_ wss 3071   ~Pcpw 3510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512
This theorem is referenced by:  pwne  4087  pwuninel2  6182
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