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Theorem dfnul2 3270
Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
Assertion
Ref Expression
dfnul2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2
StepHypRef Expression
1 df-nul 3269 . . . 4 ∅ = (V ∖ V)
21eleq2i 2149 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 2992 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 pm3.24 660 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5 eqid 2083 . . . . 5 𝑥 = 𝑥
65notnoti 607 . . . 4 ¬ ¬ 𝑥 = 𝑥
74, 62false 650 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
82, 3, 73bitri 204 . 2 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
98abbi2i 2197 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1285  wcel 1434  {cab 2069  Vcvv 2611  cdif 2980  c0 3268
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-dif 2985  df-nul 3269
This theorem is referenced by:  dfnul3  3271  rab0  3290  iotanul  4933
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