Theorem pwpw0 2460

Description: Compute the power set of the power set of the empty set. (See pw0 2459 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 2491, we have chosen to show a direct elementary proof.

Assertion

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

Proof of Theorem pwpw0

Step	Hyp	Ref	Expression
1		dfss2 2048	. . . . . . . . 9 |- (x (_ {(/)} <-> A.y(y e. x -> y e. {(/)}))
2		elsn 2411	. . . . . . . . . . 11 |- (y e. {(/)} <-> y = (/))
3	2	imbi2i 185	. . . . . . . . . 10 |- ((y e. x -> y e. {(/)}) <-> (y e. x -> y = (/)))
4	3	albii 996	. . . . . . . . 9 |- (A.y(y e. x -> y e. {(/)}) <-> A.y(y e. x -> y = (/)))
5	1, 4	bitr 173	. . . . . . . 8 |- (x (_ {(/)} <-> A.y(y e. x -> y = (/)))
6		exintr 1113	. . . . . . . . . 10 |- (A.y(y e. x -> y = (/)) -> (E.y y e. x -> E.y(y e. x /\ y = (/))))
7		n0 2279	. . . . . . . . . 10 |- (-. x = (/) <-> E.y y e. x)
8	6, 7	syl5ib 206	. . . . . . . . 9 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> E.y(y e. x /\ y = (/))))
9		exancom 1050	. . . . . . . . . . 11 |- (E.y(y e. x /\ y = (/)) <-> E.y(y = (/) /\ y e. x))
10		df-clel 1465	. . . . . . . . . . 11 |- ((/) e. x <-> E.y(y = (/) /\ y e. x))
11	9, 10	bitr4 176	. . . . . . . . . 10 |- (E.y(y e. x /\ y = (/)) <-> (/) e. x)
12		snssi 2457	. . . . . . . . . 10 |- ((/) e. x -> {(/)} (_ x)
13	11, 12	sylbi 199	. . . . . . . . 9 |- (E.y(y e. x /\ y = (/)) -> {(/)} (_ x)
14	8, 13	syl6 22	. . . . . . . 8 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> {(/)} (_ x))
15	5, 14	sylbi 199	. . . . . . 7 |- (x (_ {(/)} -> (-. x = (/) -> {(/)} (_ x))
16	15	anc2li 302	. . . . . 6 |- (x (_ {(/)} -> (-. x = (/) -> (x (_ {(/)} /\ {(/)} (_ x)))
17		eqss 2067	. . . . . 6 |- (x = {(/)} <-> (x (_ {(/)} /\ {(/)} (_ x))
18	16, 17	syl6ibr 213	. . . . 5 |- (x (_ {(/)} -> (-. x = (/) -> x = {(/)}))
19	18	orrd 233	. . . 4 |- (x (_ {(/)} -> (x = (/) \/ x = {(/)}))
20		0ss 2291	. . . . . 6 |- (/) (_ {(/)}
21		sseq1 2072	. . . . . 6 |- (x = (/) -> (x (_ {(/)} <-> (/) (_ {(/)}))
22	20, 21	mpbiri 194	. . . . 5 |- (x = (/) -> x (_ {(/)})
23		eqimss 2099	. . . . 5 |- (x = {(/)} -> x (_ {(/)})
24	22, 23	jaoi 341	. . . 4 |- ((x = (/) \/ x = {(/)}) -> x (_ {(/)})
25	19, 24	impbi 157	. . 3 |- (x (_ {(/)} <-> (x = (/) \/ x = {(/)}))
26	25	abbii 1567	. 2 |- {x | x (_ {(/)}} = {x | (x = (/) \/ x = {(/)})}
27		df-pw 2392	. 2 |- P~{(/)} = {x | x (_ {(/)}}
28		dfpr2 2412	. 2 |- {(/), {(/)}} = {x | (x = (/) \/ x = {(/)})}
29	26, 27, 28	3eqtr4 1497	1 |- P~{(/)} = {(/), {(/)}}

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  {cab 1456   (_ wss 2037  (/)c0 2270  P~cpw 2391  {csn 2399  {cpr 2400

This theorem is referenced by:  pp0ex 2761  1sdom2 4505

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403

Copyright terms: Public domain