Theorem bj-df-nul 32208

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 3878	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	bifal 1488	. 2 ⊢ (𝑥 ∈ ∅ ↔ ⊥)
3	2	bj-abbi2i 31964	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:   = wceq 1475  ⊥wfal 1480   ∈ wcel 1977  {cab 2596  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by: (None)