Theorem dmsn0el 6170

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 6166	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
2		0nelelxp 5660	. . 3 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
3	1, 2	sylbir 235	. 2 ⊢ (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴)
4	3	necon4ai 2964	1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3441  ∅c0 4286  {csn 4581   × cxp 5623  dom cdm 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-dm 5635

This theorem is referenced by:  dmsnsnsn  6179