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Theorem dfnul4 4322
Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4323. (Revised by BJ, 23-Sep-2024.)
Assertion
Ref Expression
dfnul4 ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4
StepHypRef Expression
1 df-nul 4321 . 2 ∅ = (V ∖ V)
2 df-dif 3949 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 404 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
43bifal 1558 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
54abbii 2803 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
61, 2, 53eqtri 2765 1 ∅ = {𝑥 ∣ ⊥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wfal 1554  wcel 2107  {cab 2710  Vcvv 3475  cdif 3943  c0 4320
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-dif 3949  df-nul 4321
This theorem is referenced by:  dfnul2  4323  dfnul3  4324  noel  4328  vn0  4336  eq0  4341  ab0w  4371  ab0  4372  ab0OLD  4373  abf  4400  eq0rdv  4402  rzal  4506  ralf0  4511
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