Theorem bj-dfnul2 36888

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 2152, ax-11 2168, and ax-12 2189. (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 4270	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2019	. . . 4 ⊢ 𝑥 = 𝑥
3	2	bj-ntrufal 36887	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
4	3	abbii 2807	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
5	1, 4	eqtr4i 2766	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   = wceq 1547  ⊥wfal 1559  {cab 2718  ∅c0 4268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-dif 3893  df-nul 4269

This theorem is referenced by: (None)