Theorem dfnul2OLD 4258

Description: Obsolete version of dfnul2 4256 as of 23-Sep-2024. (Contributed by NM, 26-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfnul2OLD	⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2OLD

Step	Hyp	Ref	Expression
1		df-nul 4254	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3886	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 402	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4		equid 2016	. . . . 5 ⊢ 𝑥 = 𝑥
5	4	notnoti 143	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
6	3, 5	2false 375	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
7	6	abbii 2809	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
8	1, 2, 7	3eqtri 2770	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422   ∖ cdif 3880  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-dif 3886  df-nul 4254

This theorem is referenced by: (None)