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Theorem pwpw0 4528
Description: Compute the power set of the power set of the empty set. (See pw0 4527 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4615, 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 3780 . . . . . . . . 9 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {∅}))
2 velsn 4380 . . . . . . . . . . 11 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
32imbi2i 327 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {∅}) ↔ (𝑦𝑥𝑦 = ∅))
43albii 1904 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {∅}) ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
51, 4bitri 266 . . . . . . . 8 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
6 neq0 4125 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1981 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = ∅) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = ∅)))
86, 7syl5bi 233 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = ∅)))
9 exancom 1947 . . . . . . . . . . 11 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
10 df-clel 2798 . . . . . . . . . . 11 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
119, 10bitr4i 269 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∅ ∈ 𝑥)
12 snssi 4523 . . . . . . . . . 10 (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
1311, 12sylbi 208 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = ∅) → {∅} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
155, 14sylbi 208 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
1615anc2li 547 . . . . . 6 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥)))
17 eqss 3807 . . . . . 6 (𝑥 = {∅} ↔ (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥))
1816, 17syl6ibr 243 . . . . 5 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → 𝑥 = {∅}))
1918orrd 881 . . . 4 (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))
20 0ss 4164 . . . . . 6 ∅ ⊆ {∅}
21 sseq1 3817 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {∅} ↔ ∅ ⊆ {∅}))
2220, 21mpbiri 249 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {∅})
23 eqimss 3848 . . . . 5 (𝑥 = {∅} → 𝑥 ⊆ {∅})
2422, 23jaoi 875 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ⊆ {∅})
2519, 24impbii 200 . . 3 (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
2625abbii 2919 . 2 {𝑥𝑥 ⊆ {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
27 df-pw 4347 . 2 𝒫 {∅} = {𝑥𝑥 ⊆ {∅}}
28 dfpr2 4383 . 2 {∅, {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
2926, 27, 283eqtr4i 2834 1 𝒫 {∅} = {∅, {∅}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 865  wal 1635   = wceq 1637  wex 1859  wcel 2155  {cab 2788  wss 3763  c0 4110  𝒫 cpw 4345  {csn 4364  {cpr 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-pw 4347  df-sn 4365  df-pr 4367
This theorem is referenced by:  pp0ex  5049  pwcda1  9295  canthp1lem1  9753  rankeq1o  32593  ssoninhaus  32758
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