Theorem dfnul4OLD 4359

Description: Obsolete version of dfnul4 4354 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 4355	. 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
2		equid 2011	. . . . 5 ⊢ 𝑥 = 𝑥
3	2	notnoti 143	. . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal 1553	. . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ ⊥)
5	4	abbii 2812	. 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ ⊥}
6	1, 5	eqtri 2768	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:  ¬ wn 3   = wceq 1537  ⊥wfal 1549  {cab 2717  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-dif 3979  df-nul 4353

This theorem is referenced by: (None)