Theorem dfnul4 4225

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2114, df-clel 2809. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4226. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4224	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3856	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 406	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1559	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2801	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2763	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1543  ⊥wfal 1555   ∈ wcel 2112  {cab 2714  Vcvv 3398   ∖ cdif 3850  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-dif 3856  df-nul 4224

This theorem is referenced by:  dfnul2  4226  dfnul3  4227  noel  4231  vn0  4239  eq0  4244  ab0w  4274  ab0  4275  ab0OLD  4276  abf  4303  eq0rdv  4305  rzal  4406  ralf0  4411