Theorem bj-dfnul2 36792

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 2147, ax-11 2163, and ax-12 2185. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-dfnul2

Step	Hyp	Ref	Expression
1		dfnul4 4289	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2014	. . . 4 ⊢ 𝑥 = 𝑥
3	2	bj-ntrufal 36791	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
4	3	abbii 2804	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
5	1, 4	eqtr4i 2763	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   = wceq 1542  ⊥wfal 1554  {cab 2715  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-dif 3906  df-nul 4288

This theorem is referenced by: (None)