Theorem dfnul4 4263

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2121, df-clel 2814. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4264. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4262	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3886	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 403	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1563	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2806	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2766	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547  ⊥wfal 1559   ∈ wcel 2119  {cab 2717  Vcvv 3431   ∖ cdif 3880  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-dif 3886  df-nul 4262

This theorem is referenced by:  dfnul2  4264  dfnul3  4265  noel  4266  vn0  4273  eq0  4278  ab0w  4307  ab0  4308  abf  4334  csbprc  4337  bj-dfnul2  36881