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Theorem pwsn 4398
 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 4328 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
21abbii 2736 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3 df-pw 4134 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
4 dfpr2 4168 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
52, 3, 43eqtr4i 2653 1 𝒫 {𝐴} = {∅, {𝐴}}
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 383   = wceq 1480  {cab 2607   ⊆ wss 3556  ∅c0 3893  𝒫 cpw 4132  {csn 4150  {cpr 4152 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-pw 4134  df-sn 4151  df-pr 4153 This theorem is referenced by:  pmtrsn  17863  topsn  20649  conncompid  21147  lfuhgr1v0e  26046  esumsnf  29919  cvmlift2lem9  31022  rrxtopn0b  39839  sge0sn  39919
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