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Theorem abn0 4341
Description: Nonempty class abstraction. See also ab0 4335. (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 4335 . . 3 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
21notbii 320 . 2 (¬ {𝑥𝜑} = ∅ ↔ ¬ ∀𝑥 ¬ 𝜑)
3 df-ne 2941 . 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 2940  c0 4283
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 2941  df-dif 3914  df-nul 4284
This theorem is referenced by:  intexab  5297  iinexg  5299  relimasn  6037  inisegn0  6051  mapprc  8772  modom  9191  tz9.1c  9671  scott0  9827  scott0s  9829  cp  9832  karden  9836  acnrcl  9983  aceq3lem  10061  cff  10189  cff1  10199  cfss  10206  domtriomlem  10383  axdclem  10460  nqpr  10955  supadd  12128  supmul  12132  hashf1lem2  14361  hashf1  14362  mreiincl  17481  efgval  19504  efger  19505  birthdaylem3  26319  disjex  31556  disjexc  31557  mppsval  34223  mblfinlem3  36163  ismblfin  36165  itg2addnc  36178  sdclem1  36248  upbdrech  43626
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