Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwpw0 Unicode version

Theorem pwpw0 3938
 Description: Compute the power set of the power set of the empty set. (See pw0 3937 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4001, 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 3329 . . . . . . . . 9
2 elsn 3821 . . . . . . . . . . 11
32imbi2i 304 . . . . . . . . . 10
43albii 1575 . . . . . . . . 9
51, 4bitri 241 . . . . . . . 8
6 neq0 3630 . . . . . . . . . 10
7 exintr 1624 . . . . . . . . . 10
86, 7syl5bi 209 . . . . . . . . 9
9 exancom 1596 . . . . . . . . . . 11
10 df-clel 2431 . . . . . . . . . . 11
119, 10bitr4i 244 . . . . . . . . . 10
12 snssi 3934 . . . . . . . . . 10
1311, 12sylbi 188 . . . . . . . . 9
148, 13syl6 31 . . . . . . . 8
155, 14sylbi 188 . . . . . . 7
1615anc2li 541 . . . . . 6
17 eqss 3355 . . . . . 6
1816, 17syl6ibr 219 . . . . 5
1918orrd 368 . . . 4
20 0ss 3648 . . . . . 6
21 sseq1 3361 . . . . . 6
2220, 21mpbiri 225 . . . . 5
23 eqimss 3392 . . . . 5
2422, 23jaoi 369 . . . 4
2519, 24impbii 181 . . 3
2625abbii 2547 . 2
27 df-pw 3793 . 2
28 dfpr2 3822 . 2
2926, 27, 283eqtr4i 2465 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 358   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  cab 2421   wss 3312  c0 3620  cpw 3791  csn 3806  cpr 3807 This theorem is referenced by:  pp0ex  4380  pwcda1  8063  canthp1lem1  8516  rankeq1o  26060  ssoninhaus  26146 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793  df-sn 3812  df-pr 3813
 Copyright terms: Public domain W3C validator