Theorem pwsnALT 3882

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	⊢ ℘{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 3262	. . . . . . . . 9 ⊢ (x ⊆ {A} ↔ ∀y(y ∈ x → y ∈ {A}))
2		elsn 3748	. . . . . . . . . . 11 ⊢ (y ∈ {A} ↔ y = A)
3	2	imbi2i 303	. . . . . . . . . 10 ⊢ ((y ∈ x → y ∈ {A}) ↔ (y ∈ x → y = A))
4	3	albii 1566	. . . . . . . . 9 ⊢ (∀y(y ∈ x → y ∈ {A}) ↔ ∀y(y ∈ x → y = A))
5	1, 4	bitri 240	. . . . . . . 8 ⊢ (x ⊆ {A} ↔ ∀y(y ∈ x → y = A))
6		neq0 3560	. . . . . . . . . 10 ⊢ (¬ x = ∅ ↔ ∃y y ∈ x)
7		exintr 1614	. . . . . . . . . 10 ⊢ (∀y(y ∈ x → y = A) → (∃y y ∈ x → ∃y(y ∈ x ∧ y = A)))
8	6, 7	syl5bi 208	. . . . . . . . 9 ⊢ (∀y(y ∈ x → y = A) → (¬ x = ∅ → ∃y(y ∈ x ∧ y = A)))
9		df-clel 2349	. . . . . . . . . . 11 ⊢ (A ∈ x ↔ ∃y(y = A ∧ y ∈ x))
10		exancom 1586	. . . . . . . . . . 11 ⊢ (∃y(y = A ∧ y ∈ x) ↔ ∃y(y ∈ x ∧ y = A))
11	9, 10	bitr2i 241	. . . . . . . . . 10 ⊢ (∃y(y ∈ x ∧ y = A) ↔ A ∈ x)
12		snssi 3852	. . . . . . . . . 10 ⊢ (A ∈ x → {A} ⊆ x)
13	11, 12	sylbi 187	. . . . . . . . 9 ⊢ (∃y(y ∈ x ∧ y = A) → {A} ⊆ x)
14	8, 13	syl6 29	. . . . . . . 8 ⊢ (∀y(y ∈ x → y = A) → (¬ x = ∅ → {A} ⊆ x))
15	5, 14	sylbi 187	. . . . . . 7 ⊢ (x ⊆ {A} → (¬ x = ∅ → {A} ⊆ x))
16	15	anc2li 540	. . . . . 6 ⊢ (x ⊆ {A} → (¬ x = ∅ → (x ⊆ {A} ∧ {A} ⊆ x)))
17		eqss 3287	. . . . . 6 ⊢ (x = {A} ↔ (x ⊆ {A} ∧ {A} ⊆ x))
18	16, 17	syl6ibr 218	. . . . 5 ⊢ (x ⊆ {A} → (¬ x = ∅ → x = {A}))
19	18	orrd 367	. . . 4 ⊢ (x ⊆ {A} → (x = ∅ ∨ x = {A}))
20		0ss 3579	. . . . . 6 ⊢ ∅ ⊆ {A}
21		sseq1 3292	. . . . . 6 ⊢ (x = ∅ → (x ⊆ {A} ↔ ∅ ⊆ {A}))
22	20, 21	mpbiri 224	. . . . 5 ⊢ (x = ∅ → x ⊆ {A})
23		eqimss 3323	. . . . 5 ⊢ (x = {A} → x ⊆ {A})
24	22, 23	jaoi 368	. . . 4 ⊢ ((x = ∅ ∨ x = {A}) → x ⊆ {A})
25	19, 24	impbii 180	. . 3 ⊢ (x ⊆ {A} ↔ (x = ∅ ∨ x = {A}))
26	25	abbii 2465	. 2 ⊢ {x ∣ x ⊆ {A}} = {x ∣ (x = ∅ ∨ x = {A})}
27		df-pw 3724	. 2 ⊢ ℘{A} = {x ∣ x ⊆ {A}}
28		dfpr2 3749	. 2 ⊢ {∅, {A}} = {x ∣ (x = ∅ ∨ x = {A})}
29	26, 27, 28	3eqtr4i 2383	1 ⊢ ℘{A} = {∅, {A}}

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ wss 3257  ∅c0 3550  ℘cpw 3722  {csn 3737  {cpr 3738

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742

This theorem is referenced by: (None)