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Theorem pwpw0 4312
Description: Compute the power set of the power set of the empty set. (See pw0 4311 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4396, 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.
StepHypRef Expression
1 dfss2 3572 . . . . . . . . 9 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {∅}))
2 velsn 4164 . . . . . . . . . . 11 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
32imbi2i 326 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {∅}) ↔ (𝑦𝑥𝑦 = ∅))
43albii 1744 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {∅}) ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
51, 4bitri 264 . . . . . . . 8 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
6 neq0 3906 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1816 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = ∅) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = ∅)))
86, 7syl5bi 232 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = ∅)))
9 exancom 1784 . . . . . . . . . . 11 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
10 df-clel 2617 . . . . . . . . . . 11 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
119, 10bitr4i 267 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∅ ∈ 𝑥)
12 snssi 4308 . . . . . . . . . 10 (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
1311, 12sylbi 207 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = ∅) → {∅} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
155, 14sylbi 207 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
1615anc2li 579 . . . . . 6 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥)))
17 eqss 3598 . . . . . 6 (𝑥 = {∅} ↔ (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥))
1816, 17syl6ibr 242 . . . . 5 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → 𝑥 = {∅}))
1918orrd 393 . . . 4 (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))
20 0ss 3944 . . . . . 6 ∅ ⊆ {∅}
21 sseq1 3605 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {∅} ↔ ∅ ⊆ {∅}))
2220, 21mpbiri 248 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {∅})
23 eqimss 3636 . . . . 5 (𝑥 = {∅} → 𝑥 ⊆ {∅})
2422, 23jaoi 394 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ⊆ {∅})
2519, 24impbii 199 . . 3 (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
2625abbii 2736 . 2 {𝑥𝑥 ⊆ {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
27 df-pw 4132 . 2 𝒫 {∅} = {𝑥𝑥 ⊆ {∅}}
28 dfpr2 4166 . 2 {∅, {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
2926, 27, 283eqtr4i 2653 1 𝒫 {∅} = {∅, {∅}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  {cpr 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151
This theorem is referenced by:  pp0ex  4815  pwcda1  8960  canthp1lem1  9418  rankeq1o  31917  ssoninhaus  32086
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