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Theorem pwpw0 3779
Description: Compute the power set of the power set of the empty set. (See pw0 3778 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3837, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
pwpw0  |-  ~P { (/)
}  =  { (/) ,  { (/) } }

Proof of Theorem pwpw0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3182 . . . . . . . . 9  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  e.  { (/) } ) )
2 elsn 3668 . . . . . . . . . . 11  |-  ( y  e.  { (/) }  <->  y  =  (/) )
32imbi2i 303 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { (/) } )  <->  ( y  e.  x  ->  y  =  (/) ) )
43albii 1556 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  e.  {
(/) } )  <->  A. y
( y  e.  x  ->  y  =  (/) ) )
51, 4bitri 240 . . . . . . . 8  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  =  (/) ) )
6 neq0 3478 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1604 . . . . . . . . . 10  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  y  =  (/) ) ) )
86, 7syl5bi 208 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( -.  x  =  (/)  ->  E. y
( y  e.  x  /\  y  =  (/) ) ) )
9 exancom 1576 . . . . . . . . . . 11  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
10 df-clel 2292 . . . . . . . . . . 11  |-  ( (/)  e.  x  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
119, 10bitr4i 243 . . . . . . . . . 10  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  (/)  e.  x )
12 snssi 3775 . . . . . . . . . 10  |-  ( (/)  e.  x  ->  { (/) } 
C_  x )
1311, 12sylbi 187 . . . . . . . . 9  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  ->  { (/) } 
C_  x )
148, 13syl6 29 . . . . . . . 8  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( -.  x  =  (/)  ->  { (/) } 
C_  x ) )
155, 14sylbi 187 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  { (/) }  C_  x
) )
1615anc2li 540 . . . . . 6  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  ( x  C_  { (/) }  /\  { (/) }  C_  x ) ) )
17 eqss 3207 . . . . . 6  |-  ( x  =  { (/) }  <->  ( x  C_ 
{ (/) }  /\  { (/)
}  C_  x )
)
1816, 17syl6ibr 218 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  x  =  { (/) } ) )
1918orrd 367 . . . 4  |-  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
20 0ss 3496 . . . . . 6  |-  (/)  C_  { (/) }
21 sseq1 3212 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  <->  (/)  C_  { (/) } ) )
2220, 21mpbiri 224 . . . . 5  |-  ( x  =  (/)  ->  x  C_  {
(/) } )
23 eqimss 3243 . . . . 5  |-  ( x  =  { (/) }  ->  x 
C_  { (/) } )
2422, 23jaoi 368 . . . 4  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  ->  x  C_  { (/) } )
2519, 24impbii 180 . . 3  |-  ( x 
C_  { (/) }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
2625abbii 2408 . 2  |-  { x  |  x  C_  { (/) } }  =  { x  |  ( x  =  (/)  \/  x  =  { (/)
} ) }
27 df-pw 3640 . 2  |-  ~P { (/)
}  =  { x  |  x  C_  { (/) } }
28 dfpr2 3669 . 2  |-  { (/) ,  { (/) } }  =  { x  |  (
x  =  (/)  \/  x  =  { (/) } ) }
2926, 27, 283eqtr4i 2326 1  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   {cpr 3654
This theorem is referenced by:  pp0ex  4215  pwcda1  7836  canthp1lem1  8290  rankeq1o  24873  ssoninhaus  24959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660
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