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Theorem pw0 3606
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 3341 . . 3 (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
21abbii 2210 . 2 {𝑥𝑥 ⊆ ∅} = {𝑥𝑥 = ∅}
3 df-pw 3451 . 2 𝒫 ∅ = {𝑥𝑥 ⊆ ∅}
4 df-sn 3472 . 2 {∅} = {𝑥𝑥 = ∅}
52, 3, 43eqtr4i 2125 1 𝒫 ∅ = {∅}
Colors of variables: wff set class
Syntax hints:   = wceq 1296  {cab 2081  wss 3013  c0 3302  𝒫 cpw 3449  {csn 3466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472
This theorem is referenced by:  p0ex  4044  sn0topon  11940  sn0cld  11989
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