Theorem dmsn0 6112

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 5623	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		dmsnn0 6110	. . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
3	2	necon2bbii 2995	. 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
4	1, 3	mpbir 230	1 ⊢ dom {∅} = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  {csn 4561   × cxp 5587  dom cdm 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599

This theorem is referenced by:  cnvsn0  6113  dmsnopss  6117  1st0  7837  2nd0  7838