Theorem bj-dfnul2 37051

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 2182, ax-11 2198, and ax-12 2219. (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 4296	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2039	. . . 4 ⊢ 𝑥 = 𝑥
3	2	bj-ntrufal 37050	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
4	3	abbii 2836	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
5	1, 4	eqtr4i 2795	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   = wceq 1567  ⊥wfal 1579  {cab 2747  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-dif 3916  df-nul 4295

This theorem is referenced by: (None)