Theorem dfnul3OLD 4327

Description: Obsolete version of dfnul4 4323 as of 23-Sep-2024. (Contributed by NM, 25-Mar-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfnul3OLD	⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Proof of Theorem dfnul3OLD

Step	Hyp	Ref	Expression
1		pm3.24 401	. . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)
2		equid 2013	. . . . 5 ⊢ 𝑥 = 𝑥
3	1, 2	2th 263	. . . 4 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) ↔ 𝑥 = 𝑥)
4	3	con1bii 355	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	abbii 2800	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
6		dfnul2 4324	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 3431	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
8	5, 6, 7	3eqtr4i 2768	1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Syntax hints:  ¬ wn 3   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  {crab 3430  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-rab 3431  df-dif 3950  df-nul 4322

This theorem is referenced by: (None)