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Theorem pwnss 3912
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 2102 . . . . . . 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 377 . . . . . 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 592 . . . . 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 2207 . . . . . . 7  |-  ( x  e/  x  <->  -.  x  e.  x )
5 eleq12 2102 . . . . . . . . 9  |-  ( ( x  =  y  /\  x  =  y )  ->  ( x  e.  x  <->  y  e.  y ) )
65anidms 377 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
76notbid 592 . . . . . . 7  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
84, 7syl5bb 181 . . . . . 6  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
98cbvrabv 2556 . . . . 5  |-  { x  e.  A  |  x  e/  x }  =  {
y  e.  A  |  -.  y  e.  y }
103, 9elrab2 2700 . . . 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 1265 . . . 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 2939 . . 3  |-  ( ~P A  C_  A  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  ->  { x  e.  A  |  x  e/  x }  e.  A )
)
1412, 13mtoi 590 . 2  |-  ( ~P A  C_  A  ->  -. 
{ x  e.  A  |  x  e/  x }  e.  ~P A
)
15 ssrab2 3025 . . 3  |-  { x  e.  A  |  x  e/  x }  C_  A
16 elpw2g 3910 . . 3  |-  ( A  e.  V  ->  ( { x  e.  A  |  x  e/  x }  e.  ~P A  <->  { x  e.  A  |  x  e/  x }  C_  A ) )
1715, 16mpbiri 157 . 2  |-  ( A  e.  V  ->  { x  e.  A  |  x  e/  x }  e.  ~P A )
1814, 17nsyl3 556 1  |-  ( A  e.  V  ->  -.  ~P A  C_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393    e/ wnel 2205   {crab 2310    C_ wss 2917   ~Pcpw 3359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361
This theorem is referenced by:  pwne  3913  pwuninel2  5897
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