Theorem dfnul4 4341

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2108, df-clel 2814. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4342. (Revised by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul4	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4

Step	Hyp	Ref	Expression
1		df-nul 4340	. 2 ⊢ ∅ = (V ∖ V)
2		df-dif 3966	. 2 ⊢ (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3		pm3.24 402	. . . 4 ⊢ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4	3	bifal 1553	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
5	4	abbii 2807	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
6	1, 2, 5	3eqtri 2767	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537  ⊥wfal 1549   ∈ wcel 2106  {cab 2712  Vcvv 3478   ∖ cdif 3960  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-dif 3966  df-nul 4340

This theorem is referenced by:  dfnul2  4342  dfnul3  4343  noel  4344  vn0  4351  eq0  4356  ab0w  4385  ab0  4386  abf  4412  eq0rdv  4413  rzal  4515  ralf0  4520