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Theorem dmsn0 6111
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 5624 . 2 ¬ ∅ ∈ (V × V)
2 dmsnn0 6109 . . 3 (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
32necon2bbii 2997 . 2 (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
41, 3mpbir 230 1 dom {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2110  Vcvv 3431  c0 4262  {csn 4567   × cxp 5588  dom cdm 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5596  df-dm 5600
This theorem is referenced by:  cnvsn0  6112  dmsnopss  6116  1st0  7830  2nd0  7831
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