Theorem snn0d 4741

Description: The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypothesis

Ref	Expression
snn0d.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
snn0d	⊢ (𝜑 → {𝐴} ≠ ∅)

Proof of Theorem snn0d

Step	Hyp	Ref	Expression
1		snn0d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		snnzg 4740	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2926  ∅c0 4298  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-dif 3919  df-nul 4299  df-sn 4592

This theorem is referenced by:  0nelop  5458  rnglidl0  21145  hausflim  23874  flimcf  23875  flimclslem  23877  cnpflf2  23893  cnpflf  23894  neipcfilu  24189  zarclssn  33869  zar0ring  33874  elpaddat  39793  mnuprdlem1  44254  difmapsn  45199  ovnovollem1  46647  ovnovollem3  46649