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Theorem dfnul2 4247
 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 2142, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 3-May-2023.)
Assertion
Ref Expression
dfnul2 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2
StepHypRef Expression
1 df-nul 4246 . 2 ∅ = (V ∖ V)
2 df-dif 3885 . 2 (V ∖ V) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)}
3 pm3.24 406 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
4 equid 2019 . . . . 5 𝑥 = 𝑥
54notnoti 145 . . . 4 ¬ ¬ 𝑥 = 𝑥
63, 52false 379 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
76abbii 2863 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)} = {𝑥 ∣ ¬ 𝑥 = 𝑥}
81, 2, 73eqtri 2825 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441   ∖ cdif 3879  ∅c0 4245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-dif 3885  df-nul 4246 This theorem is referenced by:  dfnul3  4248  abf  4312  iotanul  6307  avril1  28289
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