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Theorem pwpw0 4178
 Description: Compute the power set of the power set of the empty set. (See pw0 4177 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4243, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
pwpw0

Proof of Theorem pwpw0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3492 . . . . . . . . 9
2 elsn 4043 . . . . . . . . . . 11
32imbi2i 312 . . . . . . . . . 10
43albii 1640 . . . . . . . . 9
51, 4bitri 249 . . . . . . . 8
6 neq0 3795 . . . . . . . . . 10
7 exintr 1702 . . . . . . . . . 10
86, 7syl5bi 217 . . . . . . . . 9
9 exancom 1671 . . . . . . . . . . 11
10 df-clel 2452 . . . . . . . . . . 11
119, 10bitr4i 252 . . . . . . . . . 10
12 snssi 4174 . . . . . . . . . 10
1311, 12sylbi 195 . . . . . . . . 9
148, 13syl6 33 . . . . . . . 8
155, 14sylbi 195 . . . . . . 7
1615anc2li 557 . . . . . 6
17 eqss 3518 . . . . . 6
1816, 17syl6ibr 227 . . . . 5
1918orrd 378 . . . 4
20 0ss 3814 . . . . . 6
21 sseq1 3524 . . . . . 6
2220, 21mpbiri 233 . . . . 5
23 eqimss 3555 . . . . 5
2422, 23jaoi 379 . . . 4
2519, 24impbii 188 . . 3
2625abbii 2591 . 2
27 df-pw 4014 . 2
28 dfpr2 4044 . 2
2926, 27, 283eqtr4i 2496 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442  C_wss 3475   c0 3784  ~Pcpw 4012  {csn 4029  {cpr 4031 This theorem is referenced by:  pp0ex  4641  pwcda1  8595  canthp1lem1  9051  rankeq1o  29828  ssoninhaus  29913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-pw 4014  df-sn 4030  df-pr 4032
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