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Theorem elpwg 4199
 Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4857. (Contributed by NM, 6-Aug-2000.)
Assertion
Ref Expression
elpwg (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Proof of Theorem elpwg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2718 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3659 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 selpw 4198 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
41, 2, 3vtoclbg 3298 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2030   ⊆ wss 3607  𝒫 cpw 4191 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 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-in 3614  df-ss 3621  df-pw 4193 This theorem is referenced by:  elpwd  4200  elpwi  4201  elpwb  4202  pwidg  4206  elpwunsn  4256  elpwdifsn  4352  prsspwg  4387  sselpwd  4840  elpw2g  4857  pwel  4950  f1opw2  6930  eldifpw  7018  elpwun  7019  elpmg  7915  fopwdom  8109  elfpw  8309  f1opwfi  8311  rankwflemb  8694  r1elwf  8697  r1pw  8746  ackbij1lem3  9082  ackbij1lem6  9085  ackbij1lem11  9090  lcmfval  15381  lcmf0val  15382  acsfn  16367  evls1val  19733  evls1rhm  19735  evls1sca  19736  fiinopn  20754  clsval2  20902  ssntr  20910  neipeltop  20981  nrmsep3  21207  cnrmi  21212  cmpcov  21240  cmpsublem  21250  conncompss  21284  kgeni  21388  tgqtop  21563  filss  21704  ufileu  21770  filufint  21771  elutop  22084  ustuqtop0  22091  metustss  22403  psmetutop  22419  axtgcont1  25412  elpwincl1  29483  elpwdifcl  29484  elpwiuncl  29485  elpwunicl  29497  dispcmp  30054  pcmplfin  30055  indval  30203  isrnsigaOLD  30303  sigaclci  30323  sigainb  30327  elsigagen2  30339  sigapildsys  30353  ldgenpisyslem1  30354  rossros  30371  measvunilem  30403  measdivcstOLD  30415  ddeval1  30425  ddeval0  30426  omsfval  30484  omssubaddlem  30489  omssubadd  30490  elcarsg  30495  limsucncmpi  32569  topdifinffinlem  33325  elrels2  34376  ismrcd1  37578  elpwgded  39097  snelpwrVD  39380  elpwgdedVD  39467  sspwimpcf  39470  sspwimpcfVD  39471  sspwimpALT2  39478  pwpwuni  39539  wessf1ornlem  39685  dvnprodlem1  40479  dvnprodlem2  40480  ovolsplit  40523  stoweidlem50  40585  stoweidlem57  40592  pwsal  40853  saliuncl  40860  salexct  40870  fsumlesge0  40912  sge0f1o  40917  meadjuni  40992  psmeasurelem  41005  omessle  41033  caragensplit  41035  caragenelss  41036  omecl  41038  omeunile  41040  caragenuncl  41048  caragendifcl  41049  omeunle  41051  omeiunlempt  41055  carageniuncllem2  41057  carageniuncl  41058  0ome  41064  caragencmpl  41070  ovnval2  41080  ovncvrrp  41099  ovncl  41102  ovncvr2  41146  hspmbl  41164  isvonmbl  41173  smfresal  41316  gsumlsscl  42489  lincfsuppcl  42527  linccl  42528  lincdifsn  42538  lincellss  42540  ellcoellss  42549  lindslinindsimp1  42571  lindslinindimp2lem4  42575  lindslinindsimp2lem5  42576  lindslinindsimp2  42577  lincresunit3lem2  42594  lincresunit3  42595
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