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Theorem pwsn 4366
 Description: The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
Assertion
Ref Expression
pwsn 𝒫 {𝐴} = {∅, {𝐴}}

Proof of Theorem pwsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sssn 4298 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
21abbii 2726 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3 df-pw 4110 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
4 dfpr2 4143 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
52, 3, 43eqtr4i 2642 1 𝒫 {𝐴} = {∅, {𝐴}}
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   = wceq 1475  {cab 2596   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128 This theorem is referenced by:  pmtrsn  17762  topsn  20550  concompid  21044  usgra1v  25919  esumsnf  29453  cvmlift2lem9  30547  rrxtopn0b  39192  sge0sn  39272  lfuhgr1v0e  40480
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