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Theorem pwpw0 3763
Description: Compute the power set of the power set of the empty set. (See pw0 3762 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3821, 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 3169 . . . . . . . . 9  |-  ( x 
C_  { (/) }  <->  A. y
( y  e.  x  ->  y  e.  { (/) } ) )
2 elsn 3655 . . . . . . . . . . 11  |-  ( y  e.  { (/) }  <->  y  =  (/) )
32imbi2i 303 . . . . . . . . . 10  |-  ( ( y  e.  x  -> 
y  e.  { (/) } )  <->  ( y  e.  x  ->  y  =  (/) ) )
43albii 1553 . . . . . . . . 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 3465 . . . . . . . . . 10  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
7 exintr 1601 . . . . . . . . . 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 1573 . . . . . . . . . . 11  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
10 df-clel 2279 . . . . . . . . . . 11  |-  ( (/)  e.  x  <->  E. y ( y  =  (/)  /\  y  e.  x ) )
119, 10bitr4i 243 . . . . . . . . . 10  |-  ( E. y ( y  e.  x  /\  y  =  (/) )  <->  (/)  e.  x )
12 snssi 3759 . . . . . . . . . 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 3194 . . . . . 6  |-  ( x  =  { (/) }  <->  ( x  C_ 
{ (/) }  /\  { (/)
}  C_  x )
)
1816, 17syl6ibr 218 . . . . 5  |-  ( x 
C_  { (/) }  ->  ( -.  x  =  (/)  ->  x  =  { (/) } ) )
1918orrd 367 . . . 4  |-  ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) )
20 0ss 3483 . . . . . 6  |-  (/)  C_  { (/) }
21 sseq1 3199 . . . . . 6  |-  ( x  =  (/)  ->  ( x 
C_  { (/) }  <->  (/)  C_  { (/) } ) )
2220, 21mpbiri 224 . . . . 5  |-  ( x  =  (/)  ->  x  C_  {
(/) } )
23 eqimss 3230 . . . . 5  |-  ( x  =  { (/) }  ->  x 
C_  { (/) } )
2422, 23jaoi 368 . . . 4  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  ->  x  C_  { (/) } )
2519, 24impbii 180 . . 3  |-  ( x 
C_  { (/) }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
2625abbii 2395 . 2  |-  { x  |  x  C_  { (/) } }  =  { x  |  ( x  =  (/)  \/  x  =  { (/)
} ) }
27 df-pw 3627 . 2  |-  ~P { (/)
}  =  { x  |  x  C_  { (/) } }
28 dfpr2 3656 . 2  |-  { (/) ,  { (/) } }  =  { x  |  (
x  =  (/)  \/  x  =  { (/) } ) }
2926, 27, 283eqtr4i 2313 1  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   {cpr 3641
This theorem is referenced by:  pp0ex  4199  pwcda1  7820  canthp1lem1  8274  rankeq1o  24801  ssoninhaus  24887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
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