Theorem unipw 5451

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 4912	. . . 4 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴))
2		elelpwi 4613	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴)
3	2	exlimiv 1934	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴)
4	1, 3	sylbi 216	. . 3 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 → 𝑥 ∈ 𝐴)
5		vsnid 4666	. . . 4 ⊢ 𝑥 ∈ {𝑥}
6		snelpwi 5444	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
7		elunii 4914	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
8	5, 6, 7	sylancr 588	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	4, 8	impbii 208	. 2 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴)
10	9	eqriv 2730	1 ⊢ ∪ 𝒫 𝐴 = 𝐴

Syntax hints:   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  𝒫 cpw 4603  {csn 4629  ∪ cuni 4909

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  ax-sep 5300  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910

This theorem is referenced by:  univ  5452  pwtr  5453  unixpss  5811  pwexr  7752  unifpw  9355  fiuni  9423  ween  10030  fin23lem41  10347  mremre  17548  submre  17549  isacs1i  17601  eltg4i  22463  distop  22498  distopon  22500  distps  22519  ntrss2  22561  isopn3  22570  discld  22593  mretopd  22596  dishaus  22886  discmp  22902  dissnlocfin  23033  locfindis  23034  txdis  23136  xkopt  23159  xkofvcn  23188  hmphdis  23300  ustbas2  23730  vitali  25130  shsupcl  30591  shsupunss  30599  iundifdifd  31793  iundifdif  31794  dispcmp  32839  mbfmcnt  33267  omssubadd  33299  carsgval  33302  carsggect  33317  coinflipprob  33478  coinflipuniv  33480  fnemeet2  35252  bj-unirel  35932  bj-discrmoore  35992  icoreunrn  36240  ctbssinf  36287  mapdunirnN  40521  ismrcd1  41436  hbt  41872  pwelg  42311  pwsal  45031  salgenval  45037  salgenn0  45047  salexct  45050  salgencntex  45059  0ome  45245  isomennd  45247  unidmovn  45329  rrnmbl  45330  hspmbl  45345