Theorem pwpw0 2466

Description: Compute the power set of the power set of the empty set. (See pw0 2465 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 2497, we have chosen to show a direct elementary proof.

Assertion

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

Proof of Theorem pwpw0

Step	Hyp	Ref	Expression
1		dfss2 2055	. . . . . . . . 9 |- (x (_ {(/)} <-> A.y(y e. x -> y e. {(/)}))
2		elsn 2418	. . . . . . . . . . 11 |- (y e. {(/)} <-> y = (/))
3	2	imbi2i 185	. . . . . . . . . 10 |- ((y e. x -> y e. {(/)}) <-> (y e. x -> y = (/)))
4	3	albii 998	. . . . . . . . 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 1116	. . . . . . . . . 10 |- (A.y(y e. x -> y = (/)) -> (E.y y e. x -> E.y(y e. x /\ y = (/))))
7		n0 2286	. . . . . . . . . 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 1053	. . . . . . . . . . 11 |- (E.y(y e. x /\ y = (/)) <-> E.y(y = (/) /\ y e. x))
10		df-clel 1471	. . . . . . . . . . 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 2463	. . . . . . . . . 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 2074	. . . . . 6 |- (x = {(/)} <-> (x (_ {(/)} /\ {(/)} (_ x))
18	16, 17	syl6ibr 213	. . . . 5 |- (x (_ {(/)} -> (-. x = (/) -> x = {(/)}))
19	18	orrd 233	. . . 4 |- (x (_ {(/)} -> (x = (/) \/ x = {(/)}))
20		0ss 2298	. . . . . 6 |- (/) (_ {(/)}
21		sseq1 2079	. . . . . 6 |- (x = (/) -> (x (_ {(/)} <-> (/) (_ {(/)}))
22	20, 21	mpbiri 194	. . . . 5 |- (x = (/) -> x (_ {(/)})
23		eqimss 2106	. . . . 5 |- (x = {(/)} -> x (_ {(/)})
24	22, 23	jaoi 341	. . . 4 |- ((x = (/) \/ x = {(/)}) -> x (_ {(/)})
25	19, 24	impbi 157	. . 3 |- (x (_ {(/)} <-> (x = (/) \/ x = {(/)}))
26	25	abbii 1573	. 2 |- {x | x (_ {(/)}} = {x | (x = (/) \/ x = {(/)})}
27		df-pw 2399	. 2 |- P~{(/)} = {x | x (_ {(/)}}
28		dfpr2 2419	. 2 |- {(/), {(/)}} = {x | (x = (/) \/ x = {(/)})}
29	26, 27, 28	3eqtr4 1503	1 |- P~{(/)} = {(/), {(/)}}

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  {cab 1462   (_ wss 2044  (/)c0 2277  P~cpw 2398  {csn 2406  {cpr 2407

This theorem is referenced by:  pp0ex 2767  1sdom2 4514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410

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