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Theorem pwsn 4564
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 4490 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
21abbii 2887 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
3 df-pw 4297 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
4 dfpr2 4332 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
52, 3, 43eqtr4i 2802 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  wo 826   = wceq 1630  {cab 2756  wss 3721  c0 4061  𝒫 cpw 4295  {csn 4314  {cpr 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-pw 4297  df-sn 4315  df-pr 4317
This theorem is referenced by:  pmtrsn  18145  topsn  20955  conncompid  21454  lfuhgr1v0e  26368  esumsnf  30460  cvmlift2lem9  31625  rrxtopn0b  41027  sge0sn  41107
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