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Theorem pwsn 4900
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 4826 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
21abbii 2809 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3 df-pw 4602 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
4 dfpr2 4646 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
52, 3, 43eqtr4i 2775 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  wo 848   = wceq 1540  {cab 2714  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629
This theorem is referenced by:  pmtrsn  19537  topsn  22937  conncompid  23439  lfuhgr1v0e  29271  esumsnf  34065  cvmlift2lem9  35316  rrxtopn0b  46311  sge0sn  46394
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