Theorem pwpw0 1883

Description: Compute the power set of the power set of the empty set. (See pw0 1882 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48.

Assertion

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

Proof of Theorem pwpw0

Step	Hyp	Ref	Expression
1		dfss2 1497	. . . . . . . . 9 |- (x (_ {(/)} <-> A.y(y e. x -> y e. {(/)}))
2		elsn 1820	. . . . . . . . . . 11 |- (y e. {(/)} <-> y = (/))
3	2	imbi2i 160	. . . . . . . . . 10 |- ((y e. x -> y e. {(/)}) <-> (y e. x -> y = (/)))
4	3	bial 695	. . . . . . . . 9 |- (A.y(y e. x -> y e. {(/)}) <-> A.y(y e. x -> y = (/)))
5	1, 4	bitr 151	. . . . . . . 8 |- (x (_ {(/)} <-> A.y(y e. x -> y = (/)))
6		exintr 793	. . . . . . . . 9 |- (A.y(y e. x -> y = (/)) -> (E.y y e. x -> E.y(y e. x /\ y = (/))))
7		n0 1714	. . . . . . . . 9 |- (-. x = (/) <-> E.y y e. x)
8		df-clel 1099	. . . . . . . . . 10 |- ((/) e. x <-> E.y(y = (/) /\ y e. x))
9		0ex 1745	. . . . . . . . . . 11 |- (/) e. V
10	9	snss 1849	. . . . . . . . . 10 |- ((/) e. x <-> {(/)} (_ x)
11		exancom 736	. . . . . . . . . 10 |- (E.y(y = (/) /\ y e. x) <-> E.y(y e. x /\ y = (/)))
12	8, 10, 11	3bitr3 156	. . . . . . . . 9 |- ({(/)} (_ x <-> E.y(y e. x /\ y = (/)))
13	6, 7, 12	3imtr4g 426	. . . . . . . 8 |- (A.y(y e. x -> y = (/)) -> (-. x = (/) -> {(/)} (_ x))
14	5, 13	sylbi 174	. . . . . . 7 |- (x (_ {(/)} -> (-. x = (/) -> {(/)} (_ x))
15	14	anc2li 250	. . . . . 6 |- (x (_ {(/)} -> (-. x = (/) -> (x (_ {(/)} /\ {(/)} (_ x)))
16		eqss 1516	. . . . . 6 |- (x = {(/)} <-> (x (_ {(/)} /\ {(/)} (_ x))
17	15, 16	syl6ibr 186	. . . . 5 |- (x (_ {(/)} -> (-. x = (/) -> x = {(/)}))
18	17	orrd 203	. . . 4 |- (x (_ {(/)} -> (x = (/) \/ x = {(/)}))
19		0ss 1725	. . . . . 6 |- (/) (_ {(/)}
20		sseq1 1521	. . . . . 6 |- (x = (/) -> (x (_ {(/)} <-> (/) (_ {(/)}))
21	19, 20	mpbiri 169	. . . . 5 |- (x = (/) -> x (_ {(/)})
22		eqimss 1548	. . . . 5 |- (x = {(/)} -> x (_ {(/)})
23	21, 22	jaoi 275	. . . 4 |- ((x = (/) \/ x = {(/)}) -> x (_ {(/)})
24	18, 23	impbi 139	. . 3 |- (x (_ {(/)} <-> (x = (/) \/ x = {(/)}))
25	24	biabi 1181	. 2 |- {x | x (_ {(/)}} = {x | (x = (/) \/ x = {(/)})}
26		df-pw 1799	. 2 |- P~{(/)} = {x | x (_ {(/)}}
27		dfpr2 1821	. 2 |- {(/), {(/)}} = {x | (x = (/) \/ x = {(/)})}
28	25, 26, 27	3eqtr4 1126	1 |- P~{(/)} = {(/), {(/)}}

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  P~cpw 1798  {csn 1808  {cpr 1809

This theorem is referenced by:  pp0ex 1886

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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