Theorem dfnul2 4295

Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2145, ax-11 2161, and ax-12 2177. (Revised by Steven Nguyen, 3-May-2023.)

Assertion

Ref	Expression
dfnul2	⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2

Step	Hyp	Ref	Expression
1		df-nul 4294	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3941	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 405	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4		equid 2019	. . . . 5 ⊢ 𝑥 = 𝑥
5	4	notnoti 145	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
6	3, 5	2false 378	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
7	6	abbii 2888	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
8	1, 2, 7	3eqtri 2850	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  Vcvv 3496   ∖ cdif 3935  ∅c0 4293

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 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-dif 3941  df-nul 4294

This theorem is referenced by:  dfnul3  4297  iotanul  6335  avril1  28244