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Theorem dfnul2 4290
Description: Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 3-May-2023.)
Assertion
Ref Expression
dfnul2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2
StepHypRef Expression
1 df-nul 4289 . 2 ∅ = (V ∖ V)
2 df-dif 3936 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 403 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4 equid 2010 . . . . 5 𝑥 = 𝑥
54notnoti 145 . . . 4 ¬ ¬ 𝑥 = 𝑥
63, 52false 377 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
76abbii 2883 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
81, 2, 73eqtri 2845 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  cdif 3930  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-dif 3936  df-nul 4289
This theorem is referenced by:  dfnul3  4292  iotanul  6326  avril1  28169
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