Theorem dmsn0el 6187

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 6183	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		0nelelxp 5675	. . 3 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
3	1, 2	sylbir 237	. 2 ⊢ (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
4	3	necon4ai 2982	1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  Vcvv 3448  ∅c0 4280  {csn 4576   × cxp 5638  dom cdm 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646  df-dm 5650

This theorem is referenced by:  dmsnsnsn  6196