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.

Step	Hyp	Ref	Expression
1		dfss2 3492	. . . . . . . . 9
2		elsn 4043	. . . . . . . . . . 11
3	2	imbi2i 312	. . . . . . . . . 10
4	3	albii 1640	. . . . . . . . 9
5	1, 4	bitri 249	. . . . . . . 8
6		neq0 3795	. . . . . . . . . 10
7		exintr 1702	. . . . . . . . . 10
8	6, 7	syl5bi 217	. . . . . . . . 9
9		exancom 1671	. . . . . . . . . . 11
10		df-clel 2452	. . . . . . . . . . 11
11	9, 10	bitr4i 252	. . . . . . . . . 10
12		snssi 4174	. . . . . . . . . 10
13	11, 12	sylbi 195	. . . . . . . . 9
14	8, 13	syl6 33	. . . . . . . 8
15	5, 14	sylbi 195	. . . . . . 7
16	15	anc2li 557	. . . . . 6
17		eqss 3518	. . . . . 6
18	16, 17	syl6ibr 227	. . . . 5
19	18	orrd 378	. . . 4
20		0ss 3814	. . . . . 6
21		sseq1 3524	. . . . . 6
22	20, 21	mpbiri 233	. . . . 5
23		eqimss 3555	. . . . 5
24	22, 23	jaoi 379	. . . 4
25	19, 24	impbii 188	. . 3
26	25	abbii 2591	. 2
27		df-pw 4014	. 2
28		dfpr2 4044	. 2
29	26, 27, 28	3eqtr4i 2496	1

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