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Theorem dfnul2 3201
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 3200 . . . 4 ∅ = (V ∖ V)
21eleq2i 2086 . . 3 (x ∅ ↔ x (V ∖ V))
3 eldif 2902 . . 3 (x (V ∖ V) ↔ (x V ¬ x V))
4 pm3.24 614 . . . 4 ¬ (x V ¬ x V)
5 eqid 2022 . . . . 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 2134 1 ∅ = {x ∣ ¬ x = x}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1228   wcel 1374  {cab 2008  Vcvv 2533  cdif 2889  c0 3199
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-dif 2895  df-nul 3200
This theorem is referenced by:  dfnul3  3202  rab0  3221  iotanul  4807
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