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Theorem dfnul4 4335
Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2110, df-clel 2816. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4336. (Revised by BJ, 23-Sep-2024.)
Assertion
Ref Expression
dfnul4 ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4
StepHypRef Expression
1 df-nul 4334 . 2 ∅ = (V ∖ V)
2 df-dif 3954 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 402 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
43bifal 1556 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
54abbii 2809 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
61, 2, 53eqtri 2769 1 ∅ = {𝑥 ∣ ⊥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wfal 1552  wcel 2108  {cab 2714  Vcvv 3480  cdif 3948  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-dif 3954  df-nul 4334
This theorem is referenced by:  dfnul2  4336  dfnul3  4337  noel  4338  vn0  4345  eq0  4350  ab0w  4379  ab0  4380  abf  4406  eq0rdv  4407  rzal  4509  ralf0  4514
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