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Theorem pwpw0 4128
Description: Compute the power set of the power set of the empty set. (See pw0 4127 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4192, 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 3452 . . . . . . . . 9  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  e.  { (/) } ) )
2 elsn 3998 . . . . . . . . . . 11  |-  ( y  e.  { (/) }  <->  y  =  (/) )
32imbi2i 312 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { (/) } )  <->  ( y  e.  x  ->  y  =  (/) ) )
43albii 1611 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  e.  {
(/) } )  <->  A. y
( y  e.  x  ->  y  =  (/) ) )
51, 4bitri 249 . . . . . . . 8  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  =  (/) ) )
6 neq0 3754 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1669 . . . . . . . . . 10  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  y  =  (/) ) ) )
86, 7syl5bi 217 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( -.  x  =  (/)  ->  E. y
( y  e.  x  /\  y  =  (/) ) ) )
9 exancom 1639 . . . . . . . . . . 11  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
10 df-clel 2449 . . . . . . . . . . 11  |-  ( (/)  e.  x  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
119, 10bitr4i 252 . . . . . . . . . 10  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  (/)  e.  x )
12 snssi 4124 . . . . . . . . . 10  |-  ( (/)  e.  x  ->  { (/) } 
C_  x )
1311, 12sylbi 195 . . . . . . . . 9  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  ->  { (/) } 
C_  x )
148, 13syl6 33 . . . . . . . 8  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( -.  x  =  (/)  ->  { (/) } 
C_  x ) )
155, 14sylbi 195 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  { (/) }  C_  x
) )
1615anc2li 557 . . . . . 6  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  ( x  C_  { (/) }  /\  { (/) }  C_  x ) ) )
17 eqss 3478 . . . . . 6  |-  ( x  =  { (/) }  <->  ( x  C_ 
{ (/) }  /\  { (/)
}  C_  x )
)
1816, 17syl6ibr 227 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  x  =  { (/) } ) )
1918orrd 378 . . . 4  |-  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
20 0ss 3773 . . . . . 6  |-  (/)  C_  { (/) }
21 sseq1 3484 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  <->  (/)  C_  { (/) } ) )
2220, 21mpbiri 233 . . . . 5  |-  ( x  =  (/)  ->  x  C_  {
(/) } )
23 eqimss 3515 . . . . 5  |-  ( x  =  { (/) }  ->  x 
C_  { (/) } )
2422, 23jaoi 379 . . . 4  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  ->  x  C_  { (/) } )
2519, 24impbii 188 . . 3  |-  ( x 
C_  { (/) }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
2625abbii 2588 . 2  |-  { x  |  x  C_  { (/) } }  =  { x  |  ( x  =  (/)  \/  x  =  { (/)
} ) }
27 df-pw 3969 . 2  |-  ~P { (/)
}  =  { x  |  x  C_  { (/) } }
28 dfpr2 3999 . 2  |-  { (/) ,  { (/) } }  =  { x  |  (
x  =  (/)  \/  x  =  { (/) } ) }
2926, 27, 283eqtr4i 2493 1  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439    C_ wss 3435   (/)c0 3744   ~Pcpw 3967   {csn 3984   {cpr 3986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-pw 3969  df-sn 3985  df-pr 3987
This theorem is referenced by:  pp0ex  4588  pwcda1  8473  canthp1lem1  8929  rankeq1o  28352  ssoninhaus  28437
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