Theorem dmsn0el 6211

Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Assertion

Ref	Expression
dmsn0el	⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Proof of Theorem dmsn0el

Step	Hyp	Ref	Expression
1		dmsnn0 6207	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		0nelelxp 5712	. . 3 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
3	1, 2	sylbir 234	. 2 ⊢ (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
4	3	necon4ai 2973	1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  ∅c0 4323  {csn 4629   × cxp 5675  dom cdm 5677

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-dm 5687

This theorem is referenced by:  dmsnsnsn  6220