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Theorem dfnul2OLD 4175
 Description: Obsolete version of dfnul2 4174 as of 3-May-2023. Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dfnul2OLD ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Proof of Theorem dfnul2OLD
StepHypRef Expression
1 df-nul 4173 . . . 4 ∅ = (V ∖ V)
21eleq2i 2851 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 3833 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 eqid 2772 . . . . 5 𝑥 = 𝑥
5 pm3.24 394 . . . . 5 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
64, 52th 256 . . . 4 (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
76con2bii 350 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
82, 3, 73bitri 289 . 2 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
98abbi2i 2899 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2752  Vcvv 3409   ∖ cdif 3820  ∅c0 4172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-dif 3826  df-nul 4173 This theorem is referenced by: (None)
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