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Theorem dfnul2 4063
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 4062 . . . 4 ∅ = (V ∖ V)
21eleq2i 2841 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 3731 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 eqid 2770 . . . . 5 𝑥 = 𝑥
5 pm3.24 389 . . . . 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 2886 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1630  wcel 2144  {cab 2756  Vcvv 3349  cdif 3718  c0 4061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-nul 4062
This theorem is referenced by:  dfnul3  4064  iotanul  6009  avril1  27655
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