Theorem dmsn0 6197

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 5682	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		dmsnn0 6195	. . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
3	2	necon2bbii 3009	. 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
4	1, 3	mpbir 233	1 ⊢ dom {∅} = ∅

Syntax hints:  ¬ wn 3   = wceq 1561   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  {csn 4583   × cxp 5646  dom cdm 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-dm 5658

This theorem is referenced by:  cnvsn0  6198  dmsnopss  6202  1st0  7977  2nd0  7978