Theorem dfnul4 4322

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4323. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4321	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3949	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 404	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1558	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2803	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2765	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   ∧ wa 397   = wceq 1542  ⊥wfal 1554   ∈ wcel 2107  {cab 2710  Vcvv 3475   ∖ cdif 3943  ∅c0 4320

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-dif 3949  df-nul 4321

This theorem is referenced by:  dfnul2  4323  dfnul3  4324  noel  4328  vn0  4336  eq0  4341  ab0w  4371  ab0  4372  ab0OLD  4373  abf  4400  eq0rdv  4402  rzal  4506  ralf0  4511