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Theorem dfnul2 3223
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 3222 . . . 4 ∅ = (V ∖ V)
21eleq2i 2104 . . 3 (𝑥 ∈ ∅ ↔ 𝑥 ∈ (V ∖ V))
3 eldif 2924 . . 3 (𝑥 ∈ (V ∖ V) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V))
4 pm3.24 627 . . . 4 ¬ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ V)
5 eqid 2040 . . . . 5 𝑥 = 𝑥
65notnoti 574 . . . 4 ¬ ¬ 𝑥 = 𝑥
74, 62false 617 . . 3 ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ V) ↔ ¬ 𝑥 = 𝑥)
82, 3, 73bitri 195 . 2 (𝑥 ∈ ∅ ↔ ¬ 𝑥 = 𝑥)
98abbi2i 2152 1 ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97   = wceq 1243  wcel 1393  {cab 2026  Vcvv 2554  cdif 2911  c0 3221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-nul 3222
This theorem is referenced by:  dfnul3  3224  rab0  3243  iotanul  4845
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