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Theorem dmsn0 5563
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 5105 . 2 ¬ ∅ ∈ (V × V)
2 dmsnn0 5561 . . 3 (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
32necon2bbii 2841 . 2 (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
41, 3mpbir 221 1 dom {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  c0 3893  {csn 4150   × cxp 5074  dom cdm 5076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-dm 5086
This theorem is referenced by:  cnvsn0  5564  dmsnopss  5568  1st0  7122  2nd0  7123
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