Theorem pwpw0 3855

Description: Compute the power set of the power set of the empty set. (See pw0 4160 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3881, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
pwpw0	|- ~P{(/)} = {(/), {(/)}}

Proof of Theorem pwpw0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3262	. . . . . . . . 9 |- (x C_ {(/)} <-> A.y(y e. x -> y e. {(/)}))
2		elsn 3748	. . . . . . . . . . 11 |- (y e. {(/)} <-> y = (/))
3	2	imbi2i 303	. . . . . . . . . 10 |- ((y e. x -> y e. {(/)}) <-> (y e. x -> y = (/)))
4	3	albii 1566	. . . . . . . . 9 |- (A.y(y e. x -> y e. {(/)}) <-> A.y(y e. x -> y = (/)))
5	1, 4	bitri 240	. . . . . . . 8 |- (x C_ {(/)} <-> A.y(y e. x -> y = (/)))
6		neq0 3560	. . . . . . . . . 10 |- (-. x = (/) <-> E.y y e. x)
7		exintr 1614	. . . . . . . . . 10 |- (A.y(y e. x -> y = (/)) -> (E.y y e. x -> E.y(y e. x /\ y = (/))))
8	6, 7	syl5bi 208	. . . . . . . . 9 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> E.y(y e. x /\ y = (/))))
9		exancom 1586	. . . . . . . . . . 11 |- (E.y(y e. x /\ y = (/)) <-> E.y(y = (/) /\ y e. x))
10		df-clel 2349	. . . . . . . . . . 11 |- ((/) e. x <-> E.y(y = (/) /\ y e. x))
11	9, 10	bitr4i 243	. . . . . . . . . 10 |- (E.y(y e. x /\ y = (/)) <-> (/) e. x)
12		snssi 3852	. . . . . . . . . 10 |- ((/) e. x -> {(/)} C_ x)
13	11, 12	sylbi 187	. . . . . . . . 9 |- (E.y(y e. x /\ y = (/)) -> {(/)} C_ x)
14	8, 13	syl6 29	. . . . . . . 8 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> {(/)} C_ x))
15	5, 14	sylbi 187	. . . . . . 7 |- (x C_ {(/)} -> (-. x = (/) -> {(/)} C_ x))
16	15	anc2li 540	. . . . . 6 |- (x C_ {(/)} -> (-. x = (/) -> (x C_ {(/)} /\ {(/)} C_ x)))
17		eqss 3287	. . . . . 6 |- (x = {(/)} <-> (x C_ {(/)} /\ {(/)} C_ x))
18	16, 17	syl6ibr 218	. . . . 5 |- (x C_ {(/)} -> (-. x = (/) -> x = {(/)}))
19	18	orrd 367	. . . 4 |- (x C_ {(/)} -> (x = (/) \/ x = {(/)}))
20		0ss 3579	. . . . . 6 |- (/) C_ {(/)}
21		sseq1 3292	. . . . . 6 |- (x = (/) -> (x C_ {(/)} <-> (/) C_ {(/)}))
22	20, 21	mpbiri 224	. . . . 5 |- (x = (/) -> x C_ {(/)})
23		eqimss 3323	. . . . 5 |- (x = {(/)} -> x C_ {(/)})
24	22, 23	jaoi 368	. . . 4 |- ((x = (/) \/ x = {(/)}) -> x C_ {(/)})
25	19, 24	impbii 180	. . 3 |- (x C_ {(/)} <-> (x = (/) \/ x = {(/)}))
26	25	abbii 2465	. 2 |- {x | x C_ {(/)}} = {x | (x = (/) \/ x = {(/)})}
27		df-pw 3724	. 2 |- ~P{(/)} = {x | x C_ {(/)}}
28		dfpr2 3749	. 2 |- {(/), {(/)}} = {x | (x = (/) \/ x = {(/)})}
29	26, 27, 28	3eqtr4i 2383	1 |- ~P{(/)} = {(/), {(/)}}

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   C_ wss 3257  (/)c0 3550  ~Pcpw 3722  {csn 3737  {cpr 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742

This theorem is referenced by: (None)