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Theorem dfnul2 3274
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 3273 . . . 4 ∅ = (V ∖ V)
21eleq2i 2151 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 2995 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 pm3.24 660 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5 eqid 2085 . . . . 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 2199 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102   = wceq 1287  wcel 1436  {cab 2071  Vcvv 2614  cdif 2983  c0 3272
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-nul 3273
This theorem is referenced by:  dfnul3  3275  rab0  3297  iotanul  4952
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