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Theorem snelpwi 4942
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
Assertion
Ref Expression
snelpwi (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwi
StepHypRef Expression
1 snssi 4371 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 snex 4938 . . 3 {𝐴} ∈ V
32elpw 4197 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
41, 3sylibr 224 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wss 3607  𝒫 cpw 4191  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213
This theorem is referenced by:  unipw  4948  canth2  8154  unifpw  8310  marypha1lem  8380  infpwfidom  8889  ackbij1lem4  9083  acsfn  16367  sylow2a  18080  dissnref  21379  dissnlocfin  21380  locfindis  21381  txdis  21483  txdis1cn  21486  symgtgp  21952  dispcmp  30054  esumcst  30253  cntnevol  30419  coinflippvt  30674  onsucsuccmpi  32567  topdifinffinlem  33325  pclfinN  35504  lpirlnr  38004  unipwrVD  39381  unipwr  39382  salexct  40870  salexct3  40878  salgencntex  40879  salgensscntex  40880  sge0tsms  40915  sge0cl  40916  sge0sup  40926  lincvalsng  42530  snlindsntor  42585
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