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Theorem pw0 3674
 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 3406 . . 3 (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
21abbii 2256 . 2 {𝑥𝑥 ⊆ ∅} = {𝑥𝑥 = ∅}
3 df-pw 3516 . 2 𝒫 ∅ = {𝑥𝑥 ⊆ ∅}
4 df-sn 3537 . 2 {∅} = {𝑥𝑥 = ∅}
52, 3, 43eqtr4i 2171 1 𝒫 ∅ = {∅}
 Colors of variables: wff set class Syntax hints:   = wceq 1332  {cab 2126   ⊆ wss 3075  ∅c0 3367  𝒫 cpw 3514  {csn 3531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537 This theorem is referenced by:  p0ex  4119  sn0topon  12294  sn0cld  12343
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