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Theorem pwpw0 4815
Description: Compute the power set of the power set of the empty set. (See pw0 4814 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4899, 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 3967 . . . . . . . . 9 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {∅}))
2 velsn 4643 . . . . . . . . . . 11 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
32imbi2i 335 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {∅}) ↔ (𝑦𝑥𝑦 = ∅))
43albii 1819 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {∅}) ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
51, 4bitri 274 . . . . . . . 8 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
6 neq0 4344 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1893 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = ∅) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = ∅)))
86, 7biimtrid 241 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = ∅)))
9 exancom 1862 . . . . . . . . . . 11 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
10 dfclel 2809 . . . . . . . . . . 11 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
119, 10bitr4i 277 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∅ ∈ 𝑥)
12 snssi 4810 . . . . . . . . . 10 (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
1311, 12sylbi 216 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = ∅) → {∅} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
155, 14sylbi 216 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → {∅} ⊆ 𝑥))
1615anc2li 554 . . . . . 6 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥)))
17 eqss 3996 . . . . . 6 (𝑥 = {∅} ↔ (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥))
1816, 17imbitrrdi 251 . . . . 5 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → 𝑥 = {∅}))
1918orrd 859 . . . 4 (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))
20 0ss 4395 . . . . . 6 ∅ ⊆ {∅}
21 sseq1 4006 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {∅} ↔ ∅ ⊆ {∅}))
2220, 21mpbiri 257 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {∅})
23 eqimss 4039 . . . . 5 (𝑥 = {∅} → 𝑥 ⊆ {∅})
2422, 23jaoi 853 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ⊆ {∅})
2519, 24impbii 208 . . 3 (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
2625abbii 2800 . 2 {𝑥𝑥 ⊆ {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
27 df-pw 4603 . 2 𝒫 {∅} = {𝑥𝑥 ⊆ {∅}}
28 dfpr2 4646 . 2 {∅, {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
2926, 27, 283eqtr4i 2768 1 𝒫 {∅} = {∅, {∅}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 843  wal 1537   = wceq 1539  wex 1779  wcel 2104  {cab 2707  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627  {cpr 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630
This theorem is referenced by:  pp0ex  5383  pwdju1  10187  canthp1lem1  10649  rankeq1o  35447  ssoninhaus  35636
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