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Theorem dfnul2 3439
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 3438 . . . 4 ∅ = (V ∖ V)
21eleq2i 2256 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 3153 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 pm3.24 694 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5 eqid 2189 . . . . 5 𝑥 = 𝑥
65notnoti 646 . . . 4 ¬ ¬ 𝑥 = 𝑥
74, 62false 702 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
82, 3, 73bitri 206 . 2 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
98abbi2i 2304 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104   = wceq 1364  wcel 2160  {cab 2175  Vcvv 2752  cdif 3141  c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438
This theorem is referenced by:  dfnul3  3440  rab0  3466  iotanul  5211
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