Theorem pwpw0 4704

Description: Compute the power set of the power set of the empty set. (See pw0 4703 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4791, 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 3879	. . . . . . . . 9 ⊢ (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {∅}))
2		velsn 4539	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
3	2	imbi2i 340	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ {∅}) ↔ (𝑦 ∈ 𝑥 → 𝑦 = ∅))
4	3	albii 1822	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {∅}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅))
5	1, 4	bitri 278	. . . . . . . 8 ⊢ (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅))
6		neq0 4245	. . . . . . . . . 10 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		exintr 1894	. . . . . . . . . 10 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅) → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅)))
8	6, 7	syl5bi 245	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅)))
9		exancom 1863	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦 ∈ 𝑥))
10		dfclel 2832	. . . . . . . . . . 11 ⊢ (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦 ∈ 𝑥))
11	9, 10	bitr4i 281	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅) ↔ ∅ ∈ 𝑥)
12		snssi 4699	. . . . . . . . . 10 ⊢ (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
13	11, 12	sylbi 220	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = ∅) → {∅} ⊆ 𝑥)
14	8, 13	syl6 35	. . . . . . . 8 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = ∅) → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
15	5, 14	sylbi 220	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
16	15	anc2li 560	. . . . . 6 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥)))
17		eqss 3908	. . . . . 6 ⊢ (𝑥 = {∅} ↔ (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥))
18	16, 17	syl6ibr 255	. . . . 5 ⊢ (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → 𝑥 = {∅}))
19	18	orrd 861	. . . 4 ⊢ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))
20		0ss 4293	. . . . . 6 ⊢ ∅ ⊆ {∅}
21		sseq1 3918	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {∅} ↔ ∅ ⊆ {∅}))
22	20, 21	mpbiri 261	. . . . 5 ⊢ (𝑥 = ∅ → 𝑥 ⊆ {∅})
23		eqimss 3949	. . . . 5 ⊢ (𝑥 = {∅} → 𝑥 ⊆ {∅})
24	22, 23	jaoi 855	. . . 4 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ⊆ {∅})
25	19, 24	impbii 212	. . 3 ⊢ (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
26	25	abbii 2824	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
27		df-pw 4497	. 2 ⊢ 𝒫 {∅} = {𝑥 ∣ 𝑥 ⊆ {∅}}
28		dfpr2 4542	. 2 ⊢ {∅, {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
29	26, 27, 28	3eqtr4i 2792	1 ⊢ 𝒫 {∅} = {∅, {∅}}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 845  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112  {cab 2736   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  {csn 4523  {cpr 4525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-pw 4497  df-sn 4524  df-pr 4526

This theorem is referenced by:  pp0ex  5256  pwdju1  9643  canthp1lem1  10105  rankeq1o  34015  ssoninhaus  34179