Theorem dmsn0 6212

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 5697	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		dmsnn0 6210	. . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
3	2	necon2bbii 3011	. 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
4	1, 3	mpbir 234	1 ⊢ dom {∅} = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  {csn 4591   × cxp 5661  dom cdm 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-dm 5673

This theorem is used by:  cnvsn0  6213  dmsnopss  6217  1st0  7998  2nd0  7999