Theorem dfnul4 4288

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2145, df-clel 2838. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4289. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4287	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3908	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 407	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1586	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2830	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2790	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570  ⊥wfal 1582   ∈ wcel 2143  {cab 2741  Vcvv 3455   ∖ cdif 3902  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-dif 3908  df-nul 4287

This theorem is referenced by:  dfnul2  4289  dfnul3  4290  noel  4291  vn0  4298  vn0OLD  4299  eq0  4304  ab0w  4335  ab0  4336  abf  4371  csbprc  4374  bj-dfnul2  37183  bj-vn0ALT  37728