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Theorem dfnul3 3255
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 1630 . . . . 5 𝑥 = 𝑥
21notnoti 607 . . . 4 ¬ ¬ 𝑥 = 𝑥
3 pm3.24 660 . . . 4 ¬ (𝑥𝐴 ∧ ¬ 𝑥𝐴)
42, 32false 650 . . 3 𝑥 = 𝑥 ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐴))
54abbii 2195 . 2 {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
6 dfnul2 3254 . 2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
7 df-rab 2358 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐴} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐴)}
85, 6, 73eqtr4i 2112 1 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1285  wcel 1434  {cab 2068  {crab 2353  c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-dif 2976  df-nul 3253
This theorem is referenced by:  difidALT  3314
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