Theorem dmsn0 6161

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 5653	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		dmsnn0 6159	. . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅)
3	2	necon2bbii 2985	. 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V))
4	1, 3	mpbir 232	1 ⊢ dom {∅} = ∅

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4262  {csn 4556   × cxp 5617  dom cdm 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-xp 5625  df-dm 5629

This theorem is referenced by:  cnvsn0  6162  dmsnopss  6166  1st0  7938  2nd0  7939