New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  abn0 GIF version

Theorem abn0 3568
 Description: Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
abn0 ({x φ} ≠ xφ)

Proof of Theorem abn0
StepHypRef Expression
1 nfab1 2491 . . 3 x{x φ}
21n0f 3558 . 2 ({x φ} ≠ x x {x φ})
3 abid 2341 . . 3 (x {x φ} ↔ φ)
43exbii 1582 . 2 (x x {x φ} ↔ xφ)
52, 4bitri 240 1 ({x φ} ≠ xφ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∅c0 3550 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 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-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-dif 3215  df-nul 3551 This theorem is referenced by:  ab0  3569  rabn0  3570  imasn  5018  frds  5935  mapprc  6004  map0b  6024  map0  6025
 Copyright terms: Public domain W3C validator