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Theorem abn0 4381
Description: Nonempty class abstraction. See also ab0 4375. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2811, ax-8 2109. (Revised by Gino Giotto, 30-Aug-2024.)
Assertion
Ref Expression
abn0 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Proof of Theorem abn0
StepHypRef Expression
1 ab0 4375 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
21notbii 320 . 2 (¬ {𝑥𝜑} = ∅ ↔ ¬ ∀𝑥 ¬ 𝜑)
3 df-ne 2942 . 2 ({𝑥𝜑} ≠ ∅ ↔ ¬ {𝑥𝜑} = ∅)
4 df-ex 1783 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
52, 3, 43bitr4i 303 1 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1540   = wceq 1542  wex 1782  {cab 2710  wne 2941  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-ne 2942  df-dif 3952  df-nul 4324
This theorem is referenced by:  intexab  5340  iinexg  5342  relimasn  6084  inisegn0  6098  mapprc  8824  modom  9244  tz9.1c  9725  scott0  9881  scott0s  9883  cp  9886  karden  9890  acnrcl  10037  aceq3lem  10115  cff  10243  cff1  10253  cfss  10260  domtriomlem  10437  axdclem  10514  nqpr  11009  supadd  12182  supmul  12186  hashf1lem2  14417  hashf1  14418  mreiincl  17540  efgval  19585  efger  19586  birthdaylem3  26458  disjex  31823  disjexc  31824  mppsval  34563  mblfinlem3  36527  ismblfin  36529  itg2addnc  36542  sdclem1  36611  upbdrech  44015
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