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Theorem pw0 4479
 Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
pw0 𝒫 ∅ = {∅}

Proof of Theorem pw0
StepHypRef Expression
1 ss0b 4118 . . 3 (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
21abbii 2888 . 2 {𝑥𝑥 ⊆ ∅} = {𝑥𝑥 = ∅}
3 df-pw 4300 . 2 𝒫 ∅ = {𝑥𝑥 ⊆ ∅}
4 df-sn 4318 . 2 {∅} = {𝑥𝑥 = ∅}
52, 3, 43eqtr4i 2803 1 𝒫 ∅ = {∅}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631  {cab 2757   ⊆ wss 3723  ∅c0 4063  𝒫 cpw 4298  {csn 4317 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4300  df-sn 4318 This theorem is referenced by:  p0ex  4985  pwfi  8421  ackbij1lem14  9261  fin1a2lem12  9439  0tsk  9783  hashbc  13439  incexclem  14775  sn0topon  21023  sn0cld  21115  ust0  22243  uhgr0vb  26188  uhgr0  26189  esumnul  30450  rankeq1o  32615  ssoninhaus  32784  sge00  41107
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