Theorem dfnul3OLD 4329

Description: Obsolete version of dfnul4 4325 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		pm3.24 404	. . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)
2		equid 2016	. . . . 5 ⊢ 𝑥 = 𝑥
3	1, 2	2th 264	. . . 4 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) ↔ 𝑥 = 𝑥)
4	3	con1bii 357	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴))
5	4	abbii 2803	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
6		dfnul2 4326	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7		df-rab 3434	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)}
8	5, 6, 7	3eqtr4i 2771	1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}

Syntax hints:  ¬ wn 3   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3433  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-rab 3434  df-dif 3952  df-nul 4324

This theorem is referenced by: (None)