MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnul4 Structured version   Visualization version   GIF version

Theorem dfnul4 4223
Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2116, df-clel 2812. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4224. (Revised by BJ, 23-Sep-2024.)
Assertion
Ref Expression
dfnul4 ∅ = {𝑥 ∣ ⊥}

Proof of Theorem dfnul4
StepHypRef Expression
1 df-nul 4222 . 2 ∅ = (V ∖ V)
2 df-dif 3856 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 406 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
43bifal 1558 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ⊥)
54abbii 2804 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ⊥}
61, 2, 53eqtri 2766 1 ∅ = {𝑥 ∣ ⊥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1542  wfal 1554  wcel 2114  {cab 2717  Vcvv 3400  cdif 3850  c0 4221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-dif 3856  df-nul 4222
This theorem is referenced by:  dfnul2  4224  dfnul3  4225  noel  4229  vn0  4237  eq0  4242  ab0w  4272  ab0  4273  ab0OLD  4274  abf  4301  eq0rdv  4303  rzal  4405  ralf0  4410
  Copyright terms: Public domain W3C validator