Theorem pwidg 4618

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 4002	. 2 ⊢ 𝐴 ⊆ 𝐴
2		elpwg 4601	. 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 3946  𝒫 cpw 4598

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 3953  df-ss 3963  df-pw 4600

This theorem is referenced by:  pwidb  4619  pwid  4620  axpweq  5344  knatar  7341  brwdom2  9555  pwwf  9789  rankpwi  9805  canthp1lem2  10635  canthp1  10636  mremre  17535  submre  17536  baspartn  22426  fctop  22476  cctop  22478  ppttop  22479  epttop  22481  isopn3  22539  mretopd  22565  tsmsfbas  23601  gsumesum  32988  esumcst  32992  pwsiga  33059  prsiga  33060  sigainb  33065  pwldsys  33086  ldgenpisyslem1  33092  carsggect  33248  ex-sategoelel  34343  neibastop1  35149  neibastop2lem  35150  topdifinfindis  36132  elrfi  41303  dssmapnvod  42642  ntrk0kbimka  42661  clsk3nimkb  42662  neik0pk1imk0  42669  ntrclscls00  42688  ntrneicls00  42711  pwssfi  43603  dvnprodlem3  44537  caragenunidm  45097