Theorem pwidg 4623

Description: A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
pwidg	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)

Proof of Theorem pwidg

Step	Hyp	Ref	Expression
1		ssid 4005	. 2 ⊢ 𝐴 ⊆ 𝐴
2		elpwg 4606	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	1, 2	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107   ⊆ wss 3949  𝒫 cpw 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605

This theorem is referenced by:  pwidb  4624  pwid  4625  axpweq  5349  knatar  7354  brwdom2  9568  pwwf  9802  rankpwi  9818  canthp1lem2  10648  canthp1  10649  mremre  17548  submre  17549  baspartn  22457  fctop  22507  cctop  22509  ppttop  22510  epttop  22512  isopn3  22570  mretopd  22596  tsmsfbas  23632  gsumesum  33057  esumcst  33061  pwsiga  33128  prsiga  33129  sigainb  33134  pwldsys  33155  ldgenpisyslem1  33161  carsggect  33317  ex-sategoelel  34412  neibastop1  35244  neibastop2lem  35245  topdifinfindis  36227  elrfi  41432  dssmapnvod  42771  ntrk0kbimka  42790  clsk3nimkb  42791  neik0pk1imk0  42798  ntrclscls00  42817  ntrneicls00  42840  pwssfi  43732  dvnprodlem3  44664  caragenunidm  45224