Theorem bj-df-nul 34342

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-df-nul	⊢ ∅ = {𝑥 ∣ ⊥}

Proof of Theorem bj-df-nul

Step	Hyp	Ref	Expression
1		noel 4295	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	bifal 1549	. 2 ⊢ (𝑥 ∈ ∅ ↔ ⊥)
3	2	abbi2i 2953	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:   = wceq 1533  ⊥wfal 1545   ∈ wcel 2110  {cab 2799  ∅c0 4290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1536  df-fal 1546  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-dif 3938  df-nul 4291

This theorem is referenced by: (None)