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Theorem pwsn 4902
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 4831 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
21abbii 2795 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3 df-pw 4606 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
4 dfpr2 4650 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
52, 3, 43eqtr4i 2763 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  wo 845   = wceq 1533  {cab 2702  wss 3944  c0 4322  𝒫 cpw 4604  {csn 4630  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-pw 4606  df-sn 4631  df-pr 4633
This theorem is referenced by:  pmtrsn  19491  topsn  22882  conncompid  23384  lfuhgr1v0e  29144  esumsnf  33816  cvmlift2lem9  35054  rrxtopn0b  45824  sge0sn  45907
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