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Theorem pw0 3502
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 3250 . . 3 (x ⊆ ∅ ↔ x = ∅)
21abbii 2150 . 2 {xx ⊆ ∅} = {xx = ∅}
3 df-pw 3353 . 2 𝒫 ∅ = {xx ⊆ ∅}
4 df-sn 3373 . 2 {∅} = {xx = ∅}
52, 3, 43eqtr4i 2067 1 𝒫 ∅ = {∅}
Colors of variables: wff set class
Syntax hints:   = wceq 1242  {cab 2023  wss 2911  c0 3218  𝒫 cpw 3351  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373
This theorem is referenced by:  p0ex  3930
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