Theorem dfnul4 4281

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2147, df-clel 2835. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4282. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4280	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3902	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 408	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1586	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2827	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2787	1 ⊢ ∅ = {𝑥 ∣ ⊥}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ⊥wfal 1582   ∈ wcel 2145  {cab 2738  Vcvv 3450   ∖ cdif 3896  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-dif 3902  df-nul 4280

This theorem is used by:  dfnul2  4282  dfnul3  4283  noel  4284  vn0  4291  vn0OLD  4292  eq0  4297  ab0w  4328  ab0  4329  abf  4364  csbprc  4367  bj-dfnul2  37274  bj-vn0ALT  37819