Theorem snelpwi 5419

Description: If a set is a member of a class, then the singleton of that set is a member of the powerclass of that class. (Contributed by Alan Sare, 25-Aug-2011.)

Assertion

Ref	Expression
snelpwi	⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwi

Step	Hyp	Ref	Expression
1		snelpwg 5418	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
2	1	ibi 270	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  𝒫 cpw 4560  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pw 4562  df-sn 4589  df-pr 4591

This theorem is used by:  unipw  5425  canth2  9187  pwfir  9346  unifpw  9382  marypha1lem  9462  infpwfidom  10146  ackbij1lem4  10339  acsfn  17883  sylow2a  19884  dissnref  23902  dissnlocfin  23903  locfindis  23904  txdis  24006  txdis1cn  24009  symgtgp  24480  1no  28253  bday0  28254  bday0b  28256  bday1  28257  cutneg  28259  cutlt  28375  oncutlt  28707  n0bday  28795  n0fincut  28798  bdayn0p1  28812  zcuts  28850  twocut  28866  addhalfcut  28902  dispcmp  34602  esumcst  34806  cntnevol  34972  coinflippvt  35228  onsucsuccmpi  37446  topdifinffinlem  38538  pclfinN  41223  lpirlnr  44384  unipwrVD  46074  unipwr  46075  salexct  47617  salexct3  47625  salgencntex  47626  salgensscntex  47627  sge0tsms  47663  sge0cl  47664  sge0sup  47674  tmachlem-tpopen  48224  isgrtri  49314  lincvalsng  49859  snlindsntor  49914