New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  euabex GIF version

Theorem euabex 4334
 Description: If there is a unique object satisfying a property φ, then the set of all elements that satisfy φ exists. (Contributed by SF, 16-Jan-2015.)
Assertion
Ref Expression
euabex (∃!xφ → {x φ} V)

Proof of Theorem euabex
StepHypRef Expression
1 dfeu2 4333 . 2 (∃!xφ ↔ {x φ} 1c)
2 elex 2867 . 2 ({x φ} 1c → {x φ} V)
31, 2sylbi 187 1 (∃!xφ → {x φ} V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∃!weu 2204  {cab 2339  Vcvv 2859  1cc1c 4134 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136 This theorem is referenced by:  dfiota4  4372
 Copyright terms: Public domain W3C validator