Theorem dfnul2 4316

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 2142, ax-11 2158, and ax-12 2178. (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 4315	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2012	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 143	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal 1556	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
5	4	abbii 2803	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
6	1, 5	eqtr4i 2762	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   = wceq 1540  ⊥wfal 1552  {cab 2714  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-dif 3934  df-nul 4314

This theorem is referenced by:  ab0orv  4363  iotanul  6514  avril1  30449