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Theorem pwpw0 4175
 Description: Compute the power set of the power set of the empty set. (See pw0 4174 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4239, 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 3493 . . . . . . . . 9
2 elsn 4041 . . . . . . . . . . 11
32imbi2i 312 . . . . . . . . . 10
43albii 1620 . . . . . . . . 9
51, 4bitri 249 . . . . . . . 8
6 neq0 3795 . . . . . . . . . 10
7 exintr 1678 . . . . . . . . . 10
86, 7syl5bi 217 . . . . . . . . 9
9 exancom 1648 . . . . . . . . . . 11
10 df-clel 2462 . . . . . . . . . . 11
119, 10bitr4i 252 . . . . . . . . . 10
12 snssi 4171 . . . . . . . . . 10
1311, 12sylbi 195 . . . . . . . . 9
148, 13syl6 33 . . . . . . . 8
155, 14sylbi 195 . . . . . . 7
1615anc2li 557 . . . . . 6
17 eqss 3519 . . . . . 6
1816, 17syl6ibr 227 . . . . 5
1918orrd 378 . . . 4
20 0ss 3814 . . . . . 6
21 sseq1 3525 . . . . . 6
2220, 21mpbiri 233 . . . . 5
23 eqimss 3556 . . . . 5
2422, 23jaoi 379 . . . 4
2519, 24impbii 188 . . 3
2625abbii 2601 . 2
27 df-pw 4012 . 2
28 dfpr2 4042 . 2
2926, 27, 283eqtr4i 2506 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 368   wa 369  wal 1377   wceq 1379  wex 1596   wcel 1767  cab 2452   wss 3476  c0 3785  cpw 4010  csn 4027  cpr 4029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030 This theorem is referenced by:  pp0ex  4636  pwcda1  8574  canthp1lem1  9030  rankeq1o  29433  ssoninhaus  29518
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