Theorem dfnul4 4296

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2151, df-clel 2844. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4297. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4295	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3916	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 407	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1583	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2836	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2796	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567  ⊥wfal 1579   ∈ wcel 2149  {cab 2747  Vcvv 3463   ∖ cdif 3910  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-dif 3916  df-nul 4295

This theorem is referenced by:  dfnul2  4297  dfnul3  4298  noel  4299  vn0  4306  vn0OLD  4307  eq0  4312  ab0w  4342  ab0  4343  abf  4377  csbprc  4380  bj-dfnul2  37086  bj-vn0ALT  37630