Theorem pwpw0 3856

Description: Compute the power set of the power set of the empty set. (See pw0 4161 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3882, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
pwpw0	⊢ ℘{∅} = {∅, {∅}}

Proof of Theorem pwpw0

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

Step	Hyp	Ref	Expression
1		dfss2 3263	. . . . . . . . 9 ⊢ (x ⊆ {∅} ↔ ∀y(y ∈ x → y ∈ {∅}))
2		elsn 3749	. . . . . . . . . . 11 ⊢ (y ∈ {∅} ↔ y = ∅)
3	2	imbi2i 303	. . . . . . . . . 10 ⊢ ((y ∈ x → y ∈ {∅}) ↔ (y ∈ x → y = ∅))
4	3	albii 1566	. . . . . . . . 9 ⊢ (∀y(y ∈ x → y ∈ {∅}) ↔ ∀y(y ∈ x → y = ∅))
5	1, 4	bitri 240	. . . . . . . 8 ⊢ (x ⊆ {∅} ↔ ∀y(y ∈ x → y = ∅))
6		neq0 3561	. . . . . . . . . 10 ⊢ (¬ x = ∅ ↔ ∃y y ∈ x)
7		exintr 1614	. . . . . . . . . 10 ⊢ (∀y(y ∈ x → y = ∅) → (∃y y ∈ x → ∃y(y ∈ x ∧ y = ∅)))
8	6, 7	syl5bi 208	. . . . . . . . 9 ⊢ (∀y(y ∈ x → y = ∅) → (¬ x = ∅ → ∃y(y ∈ x ∧ y = ∅)))
9		exancom 1586	. . . . . . . . . . 11 ⊢ (∃y(y ∈ x ∧ y = ∅) ↔ ∃y(y = ∅ ∧ y ∈ x))
10		df-clel 2349	. . . . . . . . . . 11 ⊢ (∅ ∈ x ↔ ∃y(y = ∅ ∧ y ∈ x))
11	9, 10	bitr4i 243	. . . . . . . . . 10 ⊢ (∃y(y ∈ x ∧ y = ∅) ↔ ∅ ∈ x)
12		snssi 3853	. . . . . . . . . 10 ⊢ (∅ ∈ x → {∅} ⊆ x)
13	11, 12	sylbi 187	. . . . . . . . 9 ⊢ (∃y(y ∈ x ∧ y = ∅) → {∅} ⊆ x)
14	8, 13	syl6 29	. . . . . . . 8 ⊢ (∀y(y ∈ x → y = ∅) → (¬ x = ∅ → {∅} ⊆ x))
15	5, 14	sylbi 187	. . . . . . 7 ⊢ (x ⊆ {∅} → (¬ x = ∅ → {∅} ⊆ x))
16	15	anc2li 540	. . . . . 6 ⊢ (x ⊆ {∅} → (¬ x = ∅ → (x ⊆ {∅} ∧ {∅} ⊆ x)))
17		eqss 3288	. . . . . 6 ⊢ (x = {∅} ↔ (x ⊆ {∅} ∧ {∅} ⊆ x))
18	16, 17	syl6ibr 218	. . . . 5 ⊢ (x ⊆ {∅} → (¬ x = ∅ → x = {∅}))
19	18	orrd 367	. . . 4 ⊢ (x ⊆ {∅} → (x = ∅ ∨ x = {∅}))
20		0ss 3580	. . . . . 6 ⊢ ∅ ⊆ {∅}
21		sseq1 3293	. . . . . 6 ⊢ (x = ∅ → (x ⊆ {∅} ↔ ∅ ⊆ {∅}))
22	20, 21	mpbiri 224	. . . . 5 ⊢ (x = ∅ → x ⊆ {∅})
23		eqimss 3324	. . . . 5 ⊢ (x = {∅} → x ⊆ {∅})
24	22, 23	jaoi 368	. . . 4 ⊢ ((x = ∅ ∨ x = {∅}) → x ⊆ {∅})
25	19, 24	impbii 180	. . 3 ⊢ (x ⊆ {∅} ↔ (x = ∅ ∨ x = {∅}))
26	25	abbii 2466	. 2 ⊢ {x ∣ x ⊆ {∅}} = {x ∣ (x = ∅ ∨ x = {∅})}
27		df-pw 3725	. 2 ⊢ ℘{∅} = {x ∣ x ⊆ {∅}}
28		dfpr2 3750	. 2 ⊢ {∅, {∅}} = {x ∣ (x = ∅ ∨ x = {∅})}
29	26, 27, 28	3eqtr4i 2383	1 ⊢ ℘{∅} = {∅, {∅}}

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ wss 3258  ∅c0 3551  ℘cpw 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)