Theorem unipw 5334

Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)

Assertion

Ref	Expression
unipw	⊢ ∪ 𝒫 𝐴 = 𝐴

Proof of Theorem unipw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4834	. . . 4 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴))
2		elelpwi 4553	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴)
3	2	exlimiv 1927	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴)
4	1, 3	sylbi 219	. . 3 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 → 𝑥 ∈ 𝐴)
5		vsnid 4595	. . . 4 ⊢ 𝑥 ∈ {𝑥}
6		snelpwi 5328	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
7		elunii 4836	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
8	5, 6, 7	sylancr 589	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	4, 8	impbii 211	. 2 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴)
10	9	eqriv 2818	1 ⊢ ∪ 𝒫 𝐴 = 𝐴

Syntax hints:   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  𝒫 cpw 4538  {csn 4560  ∪ cuni 4831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561  df-pr 4563  df-uni 4832

This theorem is referenced by:  univ  5335  pwtr  5337  unixpss  5677  pwexr  7481  unifpw  8821  fiuni  8886  ween  9455  fin23lem41  9768  mremre  16869  submre  16870  isacs1i  16922  eltg4i  21562  distop  21597  distopon  21599  distps  21617  ntrss2  21659  isopn3  21668  discld  21691  mretopd  21694  dishaus  21984  discmp  22000  dissnlocfin  22131  locfindis  22132  txdis  22234  xkopt  22257  xkofvcn  22286  hmphdis  22398  ustbas2  22828  vitali  24208  shsupcl  29109  shsupunss  29117  iundifdifd  30307  iundifdif  30308  dispcmp  31118  mbfmcnt  31521  omssubadd  31553  carsgval  31556  carsggect  31571  coinflipprob  31732  coinflipuniv  31734  fnemeet2  33710  bj-unirel  34338  bj-discrmoore  34397  icoreunrn  34634  ctbssinf  34681  mapdunirnN  38780  ismrcd1  39288  hbt  39723  pwelg  39912  pwsal  42594  salgenval  42600  salgenn0  42608  salexct  42611  salgencntex  42620  0ome  42805  isomennd  42807  unidmovn  42889  rrnmbl  42890  hspmbl  42905