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Theorem dfnul3 4343
Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
Assertion
Ref Expression
dfnul3 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Proof of Theorem dfnul3
StepHypRef Expression
1 fal 1551 . . . 4 ¬ ⊥
2 pm3.24 402 . . . 4 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
31, 22false 375 . . 3 (⊥ ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
43abbii 2807 . 2 {𝑥 ∣ ⊥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
5 dfnul4 4341 . 2 ∅ = {𝑥 ∣ ⊥}
6 df-rab 3434 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
74, 5, 63eqtr4i 2773 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wfal 1549  wcel 2106  {cab 2712  {crab 3433  c0 4339
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 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-rab 3434  df-dif 3966  df-nul 4340
This theorem is referenced by:  difid  4382  kmlem3  10191
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