Theorem pwpw0 2469

Description: Compute the power set of the power set of the empty set. (See pw0 2468 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 2500, we have chosen to show a direct elementary proof.

Assertion

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

Proof of Theorem pwpw0

Step	Hyp	Ref	Expression
1		dfss2 2058	. . . . . . . . 9 |- (x (_ {(/)} <-> A.y(y e. x -> y e. {(/)}))
2		elsn 2421	. . . . . . . . . . 11 |- (y e. {(/)} <-> y = (/))
3	2	imbi2i 185	. . . . . . . . . 10 |- ((y e. x -> y e. {(/)}) <-> (y e. x -> y = (/)))
4	3	albii 999	. . . . . . . . 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 1117	. . . . . . . . . 10 |- (A.y(y e. x -> y = (/)) -> (E.y y e. x -> E.y(y e. x /\ y = (/))))
7		n0 2289	. . . . . . . . . 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 1054	. . . . . . . . . . 11 |- (E.y(y e. x /\ y = (/)) <-> E.y(y = (/) /\ y e. x))
10		df-clel 1472	. . . . . . . . . . 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 2466	. . . . . . . . . 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 2077	. . . . . 6 |- (x = {(/)} <-> (x (_ {(/)} /\ {(/)} (_ x))
18	16, 17	syl6ibr 213	. . . . 5 |- (x (_ {(/)} -> (-. x = (/) -> x = {(/)}))
19	18	orrd 233	. . . 4 |- (x (_ {(/)} -> (x = (/) \/ x = {(/)}))
20		0ss 2301	. . . . . 6 |- (/) (_ {(/)}
21		sseq1 2082	. . . . . 6 |- (x = (/) -> (x (_ {(/)} <-> (/) (_ {(/)}))
22	20, 21	mpbiri 194	. . . . 5 |- (x = (/) -> x (_ {(/)})
23		eqimss 2109	. . . . 5 |- (x = {(/)} -> x (_ {(/)})
24	22, 23	jaoi 341	. . . 4 |- ((x = (/) \/ x = {(/)}) -> x (_ {(/)})
25	19, 24	impbi 157	. . 3 |- (x (_ {(/)} <-> (x = (/) \/ x = {(/)}))
26	25	abbii 1575	. 2 |- {x | x (_ {(/)}} = {x | (x = (/) \/ x = {(/)})}
27		df-pw 2402	. 2 |- P~{(/)} = {x | x (_ {(/)}}
28		dfpr2 2422	. 2 |- {(/), {(/)}} = {x | (x = (/) \/ x = {(/)})}
29	26, 27, 28	3eqtr4 1505	1 |- P~{(/)} = {(/), {(/)}}

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   (_ wss 2047  (/)c0 2280  P~cpw 2401  {csn 2409  {cpr 2410

This theorem is referenced by:  pp0ex 2771  1sdom2 4526

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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