Theorem dfnul3 4326

Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul3	⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Proof of Theorem dfnul3

Step	Hyp	Ref	Expression
1		fal 1554	. . . 4 ⊢ ¬ ⊥
2		pm3.24 402	. . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)
3	1, 2	2false 375	. . 3 ⊢ (⊥ ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	abbii 2801	. 2 ⊢ {𝑥 ∣ ⊥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
5		dfnul4 4324	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
6		df-rab 3432	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
7	4, 5, 6	3eqtr4i 2769	1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540  ⊥wfal 1552   ∈ wcel 2105  {cab 2708  {crab 3431  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-rab 3432  df-dif 3951  df-nul 4323

This theorem is referenced by:  difid  4370  kmlem3  10153