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Theorem pw0 3507
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 3253 . . 3 (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
21abbii 2153 . 2 {𝑥𝑥 ⊆ ∅} = {𝑥𝑥 = ∅}
3 df-pw 3358 . 2 𝒫 ∅ = {𝑥𝑥 ⊆ ∅}
4 df-sn 3378 . 2 {∅} = {𝑥𝑥 = ∅}
52, 3, 43eqtr4i 2070 1 𝒫 ∅ = {∅}
Colors of variables: wff set class
Syntax hints:   = wceq 1243  {cab 2026  wss 2914  c0 3221  𝒫 cpw 3356  {csn 3372
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378
This theorem is referenced by:  p0ex  3935
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