Theorem dfnul2 4256

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 2139, ax-11 2156, and ax-12 2173. (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 4255	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2016	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 143	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal 1555	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
5	4	abbii 2809	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
6	1, 5	eqtr4i 2769	1 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Syntax hints:  ¬ wn 3   = wceq 1539  ⊥wfal 1551  {cab 2715  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-dif 3886  df-nul 4254

This theorem is referenced by:  dfnul3OLD  4259  dfnul4OLD  4260  ab0orv  4309  iotanul  6396  avril1  28728