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Theorem pwpw0 3764
Description: Compute the power set of the power set of the empty set. (See pw0 3763 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3822, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
pwpw0  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem pwpw0
StepHypRef Expression
1 dfss2 3170 . . . . . . . . 9  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  e.  { (/) } ) )
2 elsn 3656 . . . . . . . . . . 11  |-  ( y  e.  { (/) }  <->  y  =  (/) )
32imbi2i 305 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { (/) } )  <->  ( y  e.  x  ->  y  =  (/) ) )
43albii 1554 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  e.  {
(/) } )  <->  A. y
( y  e.  x  ->  y  =  (/) ) )
51, 4bitri 242 . . . . . . . 8  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  =  (/) ) )
6 neq0 3466 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1602 . . . . . . . . . 10  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  y  =  (/) ) ) )
86, 7syl5bi 210 . . . . . . . . 9  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( -.  x  =  (/)  ->  E. y
( y  e.  x  /\  y  =  (/) ) ) )
9 exancom 1574 . . . . . . . . . . 11  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
10 df-clel 2280 . . . . . . . . . . 11  |-  ( (/)  e.  x  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
119, 10bitr4i 245 . . . . . . . . . 10  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  (/)  e.  x )
12 snssi 3760 . . . . . . . . . 10  |-  ( (/)  e.  x  ->  { (/) } 
C_  x )
1311, 12sylbi 189 . . . . . . . . 9  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  ->  { (/) } 
C_  x )
148, 13syl6 31 . . . . . . . 8  |-  ( A. y ( y  e.  x  ->  y  =  (/) )  ->  ( -.  x  =  (/)  ->  { (/) } 
C_  x ) )
155, 14sylbi 189 . . . . . . 7  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  { (/) }  C_  x
) )
1615anc2li 542 . . . . . 6  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  ( x  C_  { (/) }  /\  { (/) }  C_  x ) ) )
17 eqss 3195 . . . . . 6  |-  ( x  =  { (/) }  <->  ( x  C_ 
{ (/) }  /\  { (/)
}  C_  x )
)
1816, 17syl6ibr 220 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  x  =  { (/) } ) )
1918orrd 369 . . . 4  |-  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
20 0ss 3484 . . . . . 6  |-  (/)  C_  { (/) }
21 sseq1 3200 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  <->  (/)  C_  { (/) } ) )
2220, 21mpbiri 226 . . . . 5  |-  ( x  =  (/)  ->  x  C_  {
(/) } )
23 eqimss 3231 . . . . 5  |-  ( x  =  { (/) }  ->  x 
C_  { (/) } )
2422, 23jaoi 370 . . . 4  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  ->  x  C_  { (/) } )
2519, 24impbii 182 . . 3  |-  ( x 
C_  { (/) }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
2625abbii 2396 . 2  |-  { x  |  x  C_  { (/) } }  =  { x  |  ( x  =  (/)  \/  x  =  { (/)
} ) }
27 df-pw 3628 . 2  |-  ~P { (/)
}  =  { x  |  x  C_  { (/) } }
28 dfpr2 3657 . 2  |-  { (/) ,  { (/) } }  =  { x  |  (
x  =  (/)  \/  x  =  { (/) } ) }
2926, 27, 283eqtr4i 2314 1  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   {cab 2270    C_ wss 3153   (/)c0 3456   ~Pcpw 3626   {csn 3641   {cpr 3642
This theorem is referenced by:  pp0ex  4198  pwcda1  7815  canthp1lem1  8269  rankeq1o  24208  ssoninhaus  24294
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-pw 3628  df-sn 3647  df-pr 3648
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