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Theorem dfnul4 4258
Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2108, df-clel 2816. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4259. (Revised by BJ, 23-Sep-2024.)
Assertion
Ref Expression
dfnul4 ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4
StepHypRef Expression
1 df-nul 4257 . 2 ∅ = (V ∖ V)
2 df-dif 3890 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 403 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
43bifal 1555 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
54abbii 2808 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
61, 2, 53eqtri 2770 1 ∅ = {𝑥 ∣ ⊥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wfal 1551  wcel 2106  {cab 2715  Vcvv 3432  cdif 3884  c0 4256
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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-dif 3890  df-nul 4257
This theorem is referenced by:  dfnul2  4259  dfnul3  4260  noel  4264  vn0  4272  eq0  4277  ab0w  4307  ab0  4308  ab0OLD  4309  abf  4336  eq0rdv  4338  rzal  4439  ralf0  4444
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