New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  rabn0 GIF version

Theorem rabn0 3570
 Description: Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)
Assertion
Ref Expression
rabn0 ({x A φ} ≠ x A φ)

Proof of Theorem rabn0
StepHypRef Expression
1 abn0 3568 . 2 ({x (x A φ)} ≠ x(x A φ))
2 df-rab 2623 . . 3 {x A φ} = {x (x A φ)}
32neeq1i 2526 . 2 ({x A φ} ≠ ↔ {x (x A φ)} ≠ )
4 df-rex 2620 . 2 (x A φx(x A φ))
51, 3, 43bitr4i 268 1 ({x A φ} ≠ x A φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∃wrex 2615  {crab 2618  ∅c0 3550 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-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-rex 2620  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  rabeq0  3572
 Copyright terms: Public domain W3C validator