Theorem pwpw0 4739

Description: Compute the power set of the power set of the empty set. (See pw0 4738 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4823, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)

Assertion

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

Proof of Theorem pwpw0

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

Step	Hyp	Ref	Expression
1		dfss2 3954	. . . . . . . . 9 ⊢ (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {∅}))
2		velsn 4576	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
3	2	imbi2i 338	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ {∅}) ↔ (𝑦 ∈ 𝑥 → 𝑦 = ∅))
4	3	albii 1816	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {∅}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅))
5	1, 4	bitri 277	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅))
6		neq0 4308	. . . . . . . . . 10 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		exintr 1889	. . . . . . . . . 10 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅) → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅)))
8	6, 7	syl5bi 244	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅)))
9		exancom 1857	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦 ∈ 𝑥))
10		dfclel 2894	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦 ∈ 𝑥))
11	9, 10	bitr4i 280	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅) ↔ ∅ ∈ 𝑥)
12		snssi 4734	. . . . . . . . . 10 ⊢ (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
13	11, 12	sylbi 219	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅) → {∅} ⊆ 𝑥)
14	8, 13	syl6 35	. . . . . . . 8 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅) → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
15	5, 14	sylbi 219	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
16	15	anc2li 558	. . . . . 6 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥)))
17		eqss 3981	. . . . . 6 ⊢ (𝑥 = {∅} ↔ (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥))
18	16, 17	syl6ibr 254	. . . . 5 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → 𝑥 = {∅}))
19	18	orrd 859	. . . 4 ⊢ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))
20		0ss 4349	. . . . . 6 ⊢ ∅ ⊆ {∅}
21		sseq1 3991	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {∅} ↔ ∅ ⊆ {∅}))
22	20, 21	mpbiri 260	. . . . 5 ⊢ (𝑥 = ∅ → 𝑥 ⊆ {∅})
23		eqimss 4022	. . . . 5 ⊢ (𝑥 = {∅} → 𝑥 ⊆ {∅})
24	22, 23	jaoi 853	. . . 4 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ⊆ {∅})
25	19, 24	impbii 211	. . 3 ⊢ (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
26	25	abbii 2886	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
27		df-pw 4540	. 2 ⊢ 𝒫 {∅} = {𝑥 ∣ 𝑥 ⊆ {∅}}
28		dfpr2 4579	. 2 ⊢ {∅, {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
29	26, 27, 28	3eqtr4i 2854	1 ⊢ 𝒫 {∅} = {∅, {∅}}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561  df-pr 4563

This theorem is referenced by:  pp0ex  5278  pwdju1  9610  canthp1lem1  10068  rankeq1o  33627  ssoninhaus  33791