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

Theorem euabsn 3792
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
euabsn (∃!xφx{x φ} = {x})

Proof of Theorem euabsn
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3791 . 2 (∃!xφy{x φ} = {y})
2 nfv 1619 . . 3 y{x φ} = {x}
3 nfab1 2491 . . . 4 x{x φ}
43nfeq1 2498 . . 3 x{x φ} = {y}
5 sneq 3744 . . . 4 (x = y → {x} = {y})
65eqeq2d 2364 . . 3 (x = y → ({x φ} = {x} ↔ {x φ} = {y}))
72, 4, 6cbvex 1985 . 2 (x{x φ} = {x} ↔ y{x φ} = {y})
81, 7bitr4i 243 1 (∃!xφx{x φ} = {x})
Colors of variables: wff setvar class
Syntax hints:  wb 176  wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  {csn 3737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-sn 3741
This theorem is referenced by:  eusn  3796  uniintsn  3963  mapsn  6026
  Copyright terms: Public domain W3C validator