Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnul2 Structured version   Visualization version   GIF version

Theorem dfnul2 4060
 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 4059 . . . 4 ∅ = (V ∖ V)
21eleq2i 2831 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 3725 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 eqid 2760 . . . . 5 𝑥 = 𝑥
5 pm3.24 962 . . . . 5 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
64, 52th 254 . . . 4 (𝑥 = 𝑥 ↔ ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
76con2bii 346 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
82, 3, 73bitri 286 . 2 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
98abbi2i 2876 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {cab 2746  Vcvv 3340   ∖ cdif 3712  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-nul 4059 This theorem is referenced by:  dfnul3  4061  rab0OLD  4099  iotanul  6027  avril1  27630
 Copyright terms: Public domain W3C validator