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Theorem dfnul2 3202
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 3201 . . . 4 ∅ = (V ∖ V)
21eleq2i 2087 . . 3 (x ∅ ↔ x (V ∖ V))
3 eldif 2903 . . 3 (x (V ∖ V) ↔ (x V ¬ x V))
4 pm3.24 614 . . . 4 ¬ (x V ¬ x V)
5 eqid 2023 . . . . 5 x = x
65notnoti 561 . . . 4 ¬ ¬ x = x
74, 62false 604 . . 3 ((x V ¬ x V) ↔ ¬ x = x)
82, 3, 73bitri 195 . 2 (x ∅ ↔ ¬ x = x)
98abbi2i 2135 1 ∅ = {x ∣ ¬ x = x}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1228   wcel 1375  {cab 2009  Vcvv 2534  cdif 2890  c0 3200
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-dif 2896  df-nul 3201
This theorem is referenced by:  dfnul3  3203  rab0  3222  iotanul  4807
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