Theorem bj-df-nul 34350

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 4298	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	bifal 1553	. 2 ⊢ (𝑥 ∈ ∅ ↔ ⊥)
3	2	abbi2i 2955	1 ⊢ ∅ = {𝑥 ∣ ⊥}

Syntax hints:   = wceq 1537  ⊥wfal 1549   ∈ wcel 2114  {cab 2801  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-fal 1550  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-dif 3941  df-nul 4294

This theorem is referenced by: (None)