Theorem pwsnALT 3173

Description: The power set of a singleton (direct proof).

Assertion

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

Proof of Theorem pwsnALT

Step	Hyp	Ref	Expression
1		dfss2 2610	. . . . . . . . 9 |- (x C_ {A} <-> A.y(y e. x -> y e. {A}))
2		elsn 3058	. . . . . . . . . . 11 |- (y e. {A} <-> y = A)
3	2	imbi2i 202	. . . . . . . . . 10 |- ((y e. x -> y e. {A}) <-> (y e. x -> y = A))
4	3	albii 1346	. . . . . . . . 9 |- (A.y(y e. x -> y e. {A}) <-> A.y(y e. x -> y = A))
5	1, 4	bitri 190	. . . . . . . 8 |- (x C_ {A} <-> A.y(y e. x -> y = A))
6		exintr 1475	. . . . . . . . . 10 |- (A.y(y e. x -> y = A) -> (E.y y e. x -> E.y(y e. x /\ y = A)))
7		neq0 2885	. . . . . . . . . 10 |- (-. x = (/) <-> E.y y e. x)
8	6, 7	syl5ib 223	. . . . . . . . 9 |- (A.y(y e. x -> y = A) -> (-. x = (/) -> E.y(y e. x /\ y = A)))
9		df-clel 1880	. . . . . . . . . . 11 |- (A e. x <-> E.y(y = A /\ y e. x))
10		exancom 1401	. . . . . . . . . . 11 |- (E.y(y = A /\ y e. x) <-> E.y(y e. x /\ y = A))
11	9, 10	bitr2i 191	. . . . . . . . . 10 |- (E.y(y e. x /\ y = A) <-> A e. x)
12		snssi 3129	. . . . . . . . . 10 |- (A e. x -> {A} C_ x)
13	11, 12	sylbi 216	. . . . . . . . 9 |- (E.y(y e. x /\ y = A) -> {A} C_ x)
14	8, 13	syl6 25	. . . . . . . 8 |- (A.y(y e. x -> y = A) -> (-. x = (/) -> {A} C_ x))
15	5, 14	sylbi 216	. . . . . . 7 |- (x C_ {A} -> (-. x = (/) -> {A} C_ x))
16	15	anc2li 326	. . . . . 6 |- (x C_ {A} -> (-. x = (/) -> (x C_ {A} /\ {A} C_ x)))
17		eqss 2631	. . . . . 6 |- (x = {A} <-> (x C_ {A} /\ {A} C_ x))
18	16, 17	syl6ibr 230	. . . . 5 |- (x C_ {A} -> (-. x = (/) -> x = {A}))
19	18	orrd 250	. . . 4 |- (x C_ {A} -> (x = (/) \/ x = {A}))
20		0ss 2900	. . . . . 6 |- (/) C_ {A}
21		sseq1 2637	. . . . . 6 |- (x = (/) -> (x C_ {A} <-> (/) C_ {A}))
22	20, 21	mpbiri 211	. . . . 5 |- (x = (/) -> x C_ {A})
23		eqimss 2665	. . . . 5 |- (x = {A} -> x C_ {A})
24	22, 23	jaoi 368	. . . 4 |- ((x = (/) \/ x = {A}) -> x C_ {A})
25	19, 24	impbii 174	. . 3 |- (x C_ {A} <-> (x = (/) \/ x = {A}))
26	25	abbii 2006	. 2 |- {x | x C_ {A}} = {x | (x = (/) \/ x = {A})}
27		df-pw 3035	. 2 |- ~P{A} = {x | x C_ {A}}
28		dfpr2 3059	. 2 |- {(/), {A}} = {x | (x = (/) \/ x = {A})}
29	26, 27, 28	3eqtr4i 1921	1 |- ~P{A} = {(/), {A}}

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 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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