Theorem pwsnALT 3883

Description: The power set of a singleton (direct proof). TO DO - should we keep this? (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem pwsnALT

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3263	. . . . . . . . 9 |- (x C_ {A} <-> A.y(y e. x -> y e. {A}))
2		elsn 3749	. . . . . . . . . . 11 |- (y e. {A} <-> y = A)
3	2	imbi2i 303	. . . . . . . . . 10 |- ((y e. x -> y e. {A}) <-> (y e. x -> y = A))
4	3	albii 1566	. . . . . . . . 9 |- (A.y(y e. x -> y e. {A}) <-> A.y(y e. x -> y = A))
5	1, 4	bitri 240	. . . . . . . 8 |- (x C_ {A} <-> A.y(y e. x -> y = A))
6		neq0 3561	. . . . . . . . . 10 |- (-. x = (/) <-> E.y y e. x)
7		exintr 1614	. . . . . . . . . 10 |- (A.y(y e. x -> y = A) -> (E.y y e. x -> E.y(y e. x /\ y = A)))
8	6, 7	syl5bi 208	. . . . . . . . 9 |- (A.y(y e. x -> y = A) -> (-. x = (/) -> E.y(y e. x /\ y = A)))
9		df-clel 2349	. . . . . . . . . . 11 |- (A e. x <-> E.y(y = A /\ y e. x))
10		exancom 1586	. . . . . . . . . . 11 |- (E.y(y = A /\ y e. x) <-> E.y(y e. x /\ y = A))
11	9, 10	bitr2i 241	. . . . . . . . . 10 |- (E.y(y e. x /\ y = A) <-> A e. x)
12		snssi 3853	. . . . . . . . . 10 |- (A e. x -> {A} C_ x)
13	11, 12	sylbi 187	. . . . . . . . 9 |- (E.y(y e. x /\ y = A) -> {A} C_ x)
14	8, 13	syl6 29	. . . . . . . 8 |- (A.y(y e. x -> y = A) -> (-. x = (/) -> {A} C_ x))
15	5, 14	sylbi 187	. . . . . . 7 |- (x C_ {A} -> (-. x = (/) -> {A} C_ x))
16	15	anc2li 540	. . . . . 6 |- (x C_ {A} -> (-. x = (/) -> (x C_ {A} /\ {A} C_ x)))
17		eqss 3288	. . . . . 6 |- (x = {A} <-> (x C_ {A} /\ {A} C_ x))
18	16, 17	syl6ibr 218	. . . . 5 |- (x C_ {A} -> (-. x = (/) -> x = {A}))
19	18	orrd 367	. . . 4 |- (x C_ {A} -> (x = (/) \/ x = {A}))
20		0ss 3580	. . . . . 6 |- (/) C_ {A}
21		sseq1 3293	. . . . . 6 |- (x = (/) -> (x C_ {A} <-> (/) C_ {A}))
22	20, 21	mpbiri 224	. . . . 5 |- (x = (/) -> x C_ {A})
23		eqimss 3324	. . . . 5 |- (x = {A} -> x C_ {A})
24	22, 23	jaoi 368	. . . 4 |- ((x = (/) \/ x = {A}) -> x C_ {A})
25	19, 24	impbii 180	. . 3 |- (x C_ {A} <-> (x = (/) \/ x = {A}))
26	25	abbii 2466	. 2 |- {x | x C_ {A}} = {x | (x = (/) \/ x = {A})}
27		df-pw 3725	. 2 |- ~P{A} = {x | x C_ {A}}
28		dfpr2 3750	. 2 |- {(/), {A}} = {x | (x = (/) \/ x = {A})}
29	26, 27, 28	3eqtr4i 2383	1 |- ~P{A} = {(/), {A}}

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   C_ wss 3258  (/)c0 3551  ~Pcpw 3723  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743

This theorem is referenced by: (None)