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	⊢ ℘{∅} = {∅, {∅}}

Proof of Theorem pwpw0

Step	Hyp	Ref	Expression
1		dfss2 1497	. . . . . . . . 9 ⊢ (x ⊆ {∅} ↔ ∀y(y ∈ x → y ∈ {∅}))
2		elsn 1820	. . . . . . . . . . 11 ⊢ (y ∈ {∅} ↔ y = ∅)
3	2	imbi2i 160	. . . . . . . . . 10 ⊢ ((y ∈ x → y ∈ {∅}) ↔ (y ∈ x → y = ∅))
4	3	bial 695	. . . . . . . . 9 ⊢ (∀y(y ∈ x → y ∈ {∅}) ↔ ∀y(y ∈ x → y = ∅))
5	1, 4	bitr 151	. . . . . . . 8 ⊢ (x ⊆ {∅} ↔ ∀y(y ∈ x → y = ∅))
6		exintr 793	. . . . . . . . 9 ⊢ (∀y(y ∈ x → y = ∅) → (∃y y ∈ x → ∃y(y ∈ x ∧ y = ∅)))
7		n0 1714	. . . . . . . . 9 ⊢ (¬ x = ∅ ↔ ∃y y ∈ x)
8		df-clel 1099	. . . . . . . . . 10 ⊢ (∅ ∈ x ↔ ∃y(y = ∅ ∧ y ∈ x))
9		0ex 1745	. . . . . . . . . . 11 ⊢ ∅ ∈ V
10	9	snss 1849	. . . . . . . . . 10 ⊢ (∅ ∈ x ↔ {∅} ⊆ x)
11		exancom 736	. . . . . . . . . 10 ⊢ (∃y(y = ∅ ∧ y ∈ x) ↔ ∃y(y ∈ x ∧ y = ∅))
12	8, 10, 11	3bitr3 156	. . . . . . . . 9 ⊢ ({∅} ⊆ x ↔ ∃y(y ∈ x ∧ y = ∅))
13	6, 7, 12	3imtr4g 426	. . . . . . . 8 ⊢ (∀y(y ∈ 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 ⊢ ℘{∅} = {x∣x ⊆ {∅}}
27		dfpr2 1821	. 2 ⊢ {∅, {∅}} = {x∣(x = ∅ ∨ x = {∅})}
28	25, 26, 27	3eqtr4 1126	1 ⊢ ℘{∅} = {∅, {∅}}

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ∅c0 1707  ℘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

metamath.org