Theorem dfnul2 4259

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 2137, ax-11 2154, and ax-12 2171. (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 4258	. 2 ⊢ ∅ = {𝑥 ∣ ⊥}
2		equid 2015	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 143	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal 1555	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
5	4	abbii 2808	. 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 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-dif 3890  df-nul 4257

This theorem is referenced by:  dfnul3OLD  4262  dfnul4OLD  4263  ab0orv  4312  iotanul  6411  avril1  28827