Theorem dfnul4 4283

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2148, df-clel 2839. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4284. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4282	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 4050	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 408	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1585	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2830	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2790	1 ⊢ ∅ = {𝑥 ∣ ⊥}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569  ⊥wfal 1581   ∈ wcel 2146  {cab 2741  Vcvv 3454   ∖ cdif 4048  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-dif 4050  df-nul 4282

This theorem is used by:  dfnul2  4284  dfnul3  4285  noel  4286  noelOLD  4287  vn0  4294  vn0OLD  4295  eq0  4300  ab0w  4331  ab0  4332  abf  4367  csbprc  4370  bj-dfnul2  37570  bj-vn0ALT  38129