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Theorem snelpwi 4883
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 4315 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 snex 4879 . . 3 {𝐴} ∈ V
32elpw 4142 . 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 1987  wss 3560  𝒫 cpw 4136  {csn 4155
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  ax-sep 4751  ax-nul 4759  ax-pr 4877
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 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-pw 4138  df-sn 4156  df-pr 4158
This theorem is referenced by:  unipw  4889  canth2  8073  unifpw  8229  marypha1lem  8299  infpwfidom  8811  ackbij1lem4  9005  acsfn  16260  sylow2a  17974  dissnref  21271  dissnlocfin  21272  locfindis  21273  txdis  21375  txdis1cn  21378  symgtgp  21845  dispcmp  29750  esumcst  29948  cntnevol  30114  coinflippvt  30369  onsucsuccmpi  32137  topdifinffinlem  32866  pclfinN  34705  lpirlnr  37207  unipwrVD  38589  unipwr  38590  salexct  39889  salexct3  39897  salgencntex  39898  salgensscntex  39899  sge0tsms  39934  sge0cl  39935  sge0sup  39945  lincvalsng  41523  snlindsntor  41578
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