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Theorem pwnss 3937
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 2116 . . . . . . 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 383 . . . . . 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 600 . . . . 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 2313 . . . . . . 7  |-  ( x  e/  x  <->  -.  x  e.  x )
5 eleq12 2116 . . . . . . . . 9  |-  ( ( x  =  y  /\  x  =  y )  ->  ( x  e.  x  <->  y  e.  y ) )
65anidms 383 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
76notbid 600 . . . . . . 7  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
84, 7syl5bb 185 . . . . . 6  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
98cbvrabv 2571 . . . . 5  |-  { x  e.  A  |  x  e/  x }  =  {
y  e.  A  |  -.  y  e.  y }
103, 9elrab2 2720 . . . 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 1279 . . . 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 2964 . . 3  |-  ( ~P A  C_  A  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  ->  { x  e.  A  |  x  e/  x }  e.  A )
)
1412, 13mtoi 598 . 2  |-  ( ~P A  C_  A  ->  -. 
{ x  e.  A  |  x  e/  x }  e.  ~P A
)
15 ssrab2 3050 . . 3  |-  { x  e.  A  |  x  e/  x }  C_  A
16 elpw2g 3935 . . 3  |-  ( A  e.  V  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  <->  { x  e.  A  |  x  e/  x }  C_  A ) )
1715, 16mpbiri 161 . 2  |-  ( A  e.  V  ->  { x  e.  A  |  x  e/  x }  e.  ~P A )
1814, 17nsyl3 564 1  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1257    e. wcel 1407    e/ wnel 2312   {crab 2325    C_ wss 2942   ~Pcpw 3384
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-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-nel 2313  df-rab 2330  df-v 2574  df-in 2949  df-ss 2956  df-pw 3386
This theorem is referenced by:  pwne  3938  pwuninel2  5925
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