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Theorem dfnul3 3372
 Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfnul3 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Proof of Theorem dfnul3
StepHypRef Expression
1 equid 1678 . . . . 5 𝑥 = 𝑥
21notnoti 635 . . . 4 ¬ ¬ 𝑥 = 𝑥
3 pm3.24 683 . . . 4 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
42, 32false 691 . . 3 𝑥 = 𝑥 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
54abbii 2256 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
6 dfnul2 3371 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2426 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
85, 6, 73eqtr4i 2171 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {cab 2126  {crab 2421  ∅c0 3369 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2692  df-dif 3079  df-nul 3370 This theorem is referenced by:  difidALT  3438
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