Theorem dfnul3 4285

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 1584	. . . 4 ⊢ ¬ ⊥
2		pm3.24 408	. . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)
3	1, 2	2false 378	. . 3 ⊢ (⊥ ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴))
4	3	abbii 2830	. 2 ⊢ {𝑥 ∣ ⊥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
5		dfnul4 4283	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
6		df-rab 3417	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
7	4, 5, 6	3eqtr4i 2796	1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ⊥wfal 1582   ∈ wcel 2147  {cab 2741  {crab 3416  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-rab 3417  df-dif 4050  df-nul 4282

This theorem is used by:  difid  4328  kmlem3  10270