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Theorem pwpw0 3663
 Description: Compute the power set of the power set of the empty set. (See pw0 3662 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3721, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
pwpw0

Proof of Theorem pwpw0
StepHypRef Expression
1 dfss2 3092 . . . . . . . . 9
2 elsn 3559 . . . . . . . . . . 11
32imbi2i 305 . . . . . . . . . 10
43albii 1554 . . . . . . . . 9
51, 4bitri 242 . . . . . . . 8
6 neq0 3372 . . . . . . . . . 10
7 exintr 1616 . . . . . . . . . 10
86, 7syl5bi 210 . . . . . . . . 9
9 exancom 1584 . . . . . . . . . . 11
10 df-clel 2249 . . . . . . . . . . 11
119, 10bitr4i 245 . . . . . . . . . 10
12 snssi 3659 . . . . . . . . . 10
1311, 12sylbi 189 . . . . . . . . 9
148, 13syl6 31 . . . . . . . 8
155, 14sylbi 189 . . . . . . 7
1615anc2li 542 . . . . . 6
17 eqss 3115 . . . . . 6
1816, 17syl6ibr 220 . . . . 5
1918orrd 369 . . . 4
20 0ss 3390 . . . . . 6
21 sseq1 3120 . . . . . 6
2220, 21mpbiri 226 . . . . 5
23 eqimss 3151 . . . . 5
2422, 23jaoi 370 . . . 4
2519, 24impbii 182 . . 3
2625abbii 2361 . 2
27 df-pw 3532 . 2
28 dfpr2 3560 . 2
2926, 27, 283eqtr4i 2283 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wo 359   wa 360  wal 1532  wex 1537   wceq 1619   wcel 1621  cab 2239   wss 3078  c0 3362  cpw 3530  csn 3544  cpr 3545 This theorem is referenced by:  pp0ex  4093  pwcda1  7704  canthp1lem1  8154  rankeq1o  23975  ssoninhaus  24061 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-pw 3532  df-sn 3550  df-pr 3551
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