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Theorem dfnul2 3552
 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 3551 . . . 4 = (V V)
21eleq2i 2417 . . 3 (x x (V V))
3 eldif 3221 . . 3 (x (V V) ↔ (x V ¬ x V))
4 eqid 2353 . . . . 5 x = x
5 pm3.24 852 . . . . 5 ¬ (x V ¬ x V)
64, 52th 230 . . . 4 (x = x ↔ ¬ (x V ¬ x V))
76con2bii 322 . . 3 ((x V ¬ x V) ↔ ¬ x = x)
82, 3, 73bitri 262 . 2 (x ↔ ¬ x = x)
98abbi2i 2464 1 = {x ¬ x = x}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859   ∖ cdif 3206  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  dfnul3  3553  rab0  3571  iotanul  4354
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