Theorem pwpw0 3134

Description: Compute the power set of the power set of the empty set. (See pw0 3132 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3172, we have chosen to show a direct elementary proof.

Assertion

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

Proof of Theorem pwpw0

Step	Hyp	Ref	Expression
1		dfss2 2610	. . . . . . . . 9 |- (x C_ {(/)} <-> A.y(y e. x -> y e. {(/)}))
2		elsn 3058	. . . . . . . . . . 11 |- (y e. {(/)} <-> y = (/))
3	2	imbi2i 202	. . . . . . . . . 10 |- ((y e. x -> y e. {(/)}) <-> (y e. x -> y = (/)))
4	3	albii 1346	. . . . . . . . 9 |- (A.y(y e. x -> y e. {(/)}) <-> A.y(y e. x -> y = (/)))
5	1, 4	bitri 190	. . . . . . . 8 |- (x C_ {(/)} <-> A.y(y e. x -> y = (/)))
6		exintr 1475	. . . . . . . . . 10 |- (A.y(y e. x -> y = (/)) -> (E.y y e. x -> E.y(y e. x /\ y = (/))))
7		neq0 2885	. . . . . . . . . 10 |- (-. x = (/) <-> E.y y e. x)
8	6, 7	syl5ib 223	. . . . . . . . 9 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> E.y(y e. x /\ y = (/))))
9		exancom 1401	. . . . . . . . . . 11 |- (E.y(y e. x /\ y = (/)) <-> E.y(y = (/) /\ y e. x))
10		df-clel 1880	. . . . . . . . . . 11 |- ((/) e. x <-> E.y(y = (/) /\ y e. x))
11	9, 10	bitr4i 193	. . . . . . . . . 10 |- (E.y(y e. x /\ y = (/)) <-> (/) e. x)
12		snssi 3129	. . . . . . . . . 10 |- ((/) e. x -> {(/)} C_ x)
13	11, 12	sylbi 216	. . . . . . . . 9 |- (E.y(y e. x /\ y = (/)) -> {(/)} C_ x)
14	8, 13	syl6 25	. . . . . . . 8 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> {(/)} C_ x))
15	5, 14	sylbi 216	. . . . . . 7 |- (x C_ {(/)} -> (-. x = (/) -> {(/)} C_ x))
16	15	anc2li 326	. . . . . 6 |- (x C_ {(/)} -> (-. x = (/) -> (x C_ {(/)} /\ {(/)} C_ x)))
17		eqss 2631	. . . . . 6 |- (x = {(/)} <-> (x C_ {(/)} /\ {(/)} C_ x))
18	16, 17	syl6ibr 230	. . . . 5 |- (x C_ {(/)} -> (-. x = (/) -> x = {(/)}))
19	18	orrd 250	. . . 4 |- (x C_ {(/)} -> (x = (/) \/ x = {(/)}))
20		0ss 2900	. . . . . 6 |- (/) C_ {(/)}
21		sseq1 2637	. . . . . 6 |- (x = (/) -> (x C_ {(/)} <-> (/) C_ {(/)}))
22	20, 21	mpbiri 211	. . . . 5 |- (x = (/) -> x C_ {(/)})
23		eqimss 2665	. . . . 5 |- (x = {(/)} -> x C_ {(/)})
24	22, 23	jaoi 368	. . . 4 |- ((x = (/) \/ x = {(/)}) -> x C_ {(/)})
25	19, 24	impbii 174	. . 3 |- (x C_ {(/)} <-> (x = (/) \/ x = {(/)}))
26	25	abbii 2006	. 2 |- {x | x C_ {(/)}} = {x | (x = (/) \/ x = {(/)})}
27		df-pw 3035	. 2 |- ~P{(/)} = {x | x C_ {(/)}}
28		dfpr2 3059	. 2 |- {(/), {(/)}} = {x | (x = (/) \/ x = {(/)})}
29	26, 27, 28	3eqtr4i 1921	1 |- ~P{(/)} = {(/), {(/)}}

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044  {cpr 3045

This theorem is referenced by:  pp0ex 3496  1sdom2 5619  tartwo 15233

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050

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