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Theorem dmsn0 5856
 Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
Assertion
Ref Expression
dmsn0 dom {∅} = ∅

Proof of Theorem dmsn0
StepHypRef Expression
1 0nelxp 5389 . 2 ¬ ∅ ∈ (V × V)
2 dmsnn0 5854 . . 3 (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
32necon2bbii 3019 . 2 (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
41, 3mpbir 223 1 dom {∅} = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1601   ∈ wcel 2106  Vcvv 3397  ∅c0 4140  {csn 4397   × cxp 5353  dom cdm 5355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-dm 5365 This theorem is referenced by:  cnvsn0  5857  dmsnopss  5861  1st0  7451  2nd0  7452
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