Theorem dfnul2 4317

Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2129, ax-11 2146, and ax-12 2163. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)

Assertion

Ref	Expression
dfnul2	⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2

Step	Hyp	Ref	Expression
1		dfnul4 4316	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2007	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 143	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal 1549	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
5	4	abbii 2794	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
6	1, 5	eqtr4i 2755	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   = wceq 1533  ⊥wfal 1545  {cab 2701  ∅c0 4314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-dif 3943  df-nul 4315

This theorem is referenced by:  dfnul3OLD  4320  dfnul4OLD  4321  ab0orv  4370  iotanul  6511  avril1  30185