Theorem snn0d 4800

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 4799	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-dif 3979  df-nul 4353  df-sn 4649

This theorem is referenced by:  0nelop  5515  rnglidl0  21262  hausflim  24010  flimcf  24011  flimclslem  24013  cnpflf2  24029  cnpflf  24030  neipcfilu  24326  zarclssn  33819  zar0ring  33824  elpaddat  39761  mnuprdlem1  44241  difmapsn  45119  ovnovollem1  46577  ovnovollem3  46579