ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  unipw Structured version   GIF version

Theorem unipw 3944
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.
StepHypRef Expression
1 eluni 3574 . . . 4 (x 𝒫 Ay(x y y 𝒫 A))
2 elelpwi 3362 . . . . 5 ((x y y 𝒫 A) → x A)
32exlimiv 1486 . . . 4 (y(x y y 𝒫 A) → x A)
41, 3sylbi 114 . . 3 (x 𝒫 Ax A)
5 vex 2554 . . . . 5 x V
65snid 3394 . . . 4 x {x}
7 snelpwi 3939 . . . 4 (x A → {x} 𝒫 A)
8 elunii 3576 . . . 4 ((x {x} {x} 𝒫 A) → x 𝒫 A)
96, 7, 8sylancr 393 . . 3 (x Ax 𝒫 A)
104, 9impbii 117 . 2 (x 𝒫 Ax A)
1110eqriv 2034 1 𝒫 A = A
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  𝒫 cpw 3351  {csn 3367   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-uni 3572
This theorem is referenced by:  pwtr  3946  pwexb  4172  univ  4173  unixpss  4394
  Copyright terms: Public domain W3C validator