Theorem unipw 5389

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 4859	. . . 4 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴))
2		elelpwi 4557	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴)
3	2	exlimiv 1931	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝐴)
4	1, 3	sylbi 217	. . 3 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 → 𝑥 ∈ 𝐴)
5		vsnid 4613	. . . 4 ⊢ 𝑥 ∈ {𝑥}
6		snelpwi 5383	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
7		elunii 4861	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
8	5, 6, 7	sylancr 587	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	4, 8	impbii 209	. 2 ⊢ (𝑥 ∈ ∪ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴)
10	9	eqriv 2728	1 ⊢ ∪ 𝒫 𝐴 = 𝐴

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  𝒫 cpw 4547  {csn 4573  ∪ cuni 4856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-pw 4549  df-sn 4574  df-pr 4576  df-uni 4857

This theorem is referenced by:  univ  5390  pwtr  5391  unixpss  5749  pwexr  7698  unifpw  9239  fiuni  9312  ween  9926  fin23lem41  10243  mremre  17506  submre  17507  isacs1i  17563  eltg4i  22875  distop  22910  distopon  22912  distps  22930  ntrss2  22972  isopn3  22981  discld  23004  mretopd  23007  dishaus  23297  discmp  23313  dissnlocfin  23444  locfindis  23445  txdis  23547  xkopt  23570  xkofvcn  23599  hmphdis  23711  ustbas2  24140  vitali  25541  shsupcl  31318  shsupunss  31326  iundifdifd  32541  iundifdif  32542  dispcmp  33872  mbfmcnt  34281  omssubadd  34313  carsgval  34316  carsggect  34331  coinflipprob  34493  coinflipuniv  34495  fnemeet2  36409  bj-unirel  37093  bj-discrmoore  37153  icoreunrn  37401  ctbssinf  37448  mapdunirnN  41697  ismrcd1  42739  hbt  43171  pwelg  43601  pwsal  46361  salgenval  46367  salgenn0  46377  salexct  46380  salgencntex  46389  0ome  46575  isomennd  46577  unidmovn  46659  rrnmbl  46660  hspmbl  46675