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Theorem pw0 3561
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 3307 . . 3 (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
21abbii 2200 . 2 {𝑥𝑥 ⊆ ∅} = {𝑥𝑥 = ∅}
3 df-pw 3411 . 2 𝒫 ∅ = {𝑥𝑥 ⊆ ∅}
4 df-sn 3431 . 2 {∅} = {𝑥𝑥 = ∅}
52, 3, 43eqtr4i 2115 1 𝒫 ∅ = {∅}
Colors of variables: wff set class
Syntax hints:   = wceq 1287  {cab 2071  wss 2986  c0 3272  𝒫 cpw 3409  {csn 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431
This theorem is referenced by:  p0ex  3990
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