Theorem dmsn0 6165

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

Step	Hyp	Ref	Expression
1		0nelxp 5656	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		dmsnn0 6163	. . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
3	2	necon2bbii 2981	. 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
4	1, 3	mpbir 231	1 ⊢ dom {∅} = ∅

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  {csn 4578   × cxp 5620  dom cdm 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-dm 5632

This theorem is referenced by:  cnvsn0  6166  dmsnopss  6170  1st0  7937  2nd0  7938