Theorem dfnul4OLD 4230

Description: Obsolete version of dfnul4 4225 as of 23-Sep-2024. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfnul4OLD	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4OLD

Step	Hyp	Ref	Expression
1		dfnul2 4226	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
2		equid 2022	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 145	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal 1559	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
5	4	abbii 2801	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
6	1, 5	eqtri 2759	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   = wceq 1543  ⊥wfal 1555  {cab 2714  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-dif 3856  df-nul 4224

This theorem is referenced by: (None)