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Theorem pwpw0 4477
Description: Compute the power set of the power set of the empty set. (See pw0 4476 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4568, 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 3720 . . . . . . . . 9 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {∅}))
2 velsn 4325 . . . . . . . . . . 11 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
32imbi2i 325 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {∅}) ↔ (𝑦𝑥𝑦 = ∅))
43albii 1884 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {∅}) ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
51, 4bitri 264 . . . . . . . 8 (𝑥 ⊆ {∅} ↔ ∀𝑦(𝑦𝑥𝑦 = ∅))
6 neq0 4061 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1956 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = ∅) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = ∅)))
86, 7syl5bi 232 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = ∅) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = ∅)))
9 exancom 1924 . . . . . . . . . . 11 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
10 df-clel 2744 . . . . . . . . . . 11 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑦𝑥))
119, 10bitr4i 267 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = ∅) ↔ ∅ ∈ 𝑥)
12 snssi 4472 . . . . . . . . . 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 581 . . . . . 6 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥)))
17 eqss 3747 . . . . . 6 (𝑥 = {∅} ↔ (𝑥 ⊆ {∅} ∧ {∅} ⊆ 𝑥))
1816, 17syl6ibr 242 . . . . 5 (𝑥 ⊆ {∅} → (¬ 𝑥 = ∅ → 𝑥 = {∅}))
1918orrd 392 . . . 4 (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅}))
20 0ss 4103 . . . . . 6 ∅ ⊆ {∅}
21 sseq1 3755 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {∅} ↔ ∅ ⊆ {∅}))
2220, 21mpbiri 248 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {∅})
23 eqimss 3786 . . . . 5 (𝑥 = {∅} → 𝑥 ⊆ {∅})
2422, 23jaoi 393 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {∅}) → 𝑥 ⊆ {∅})
2519, 24impbii 199 . . 3 (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
2625abbii 2865 . 2 {𝑥𝑥 ⊆ {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
27 df-pw 4292 . 2 𝒫 {∅} = {𝑥𝑥 ⊆ {∅}}
28 dfpr2 4327 . 2 {∅, {∅}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {∅})}
2926, 27, 283eqtr4i 2780 1 𝒫 {∅} = {∅, {∅}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  wal 1618   = wceq 1620  wex 1841  wcel 2127  {cab 2734  wss 3703  c0 4046  𝒫 cpw 4290  {csn 4309  {cpr 4311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-pw 4292  df-sn 4310  df-pr 4312
This theorem is referenced by:  pp0ex  4992  pwcda1  9179  canthp1lem1  9637  rankeq1o  32555  ssoninhaus  32724
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