Theorem snn0d 4734

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 4733	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ∅c0 4287  {csn 4582

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

This theorem depends on definitions:  df-bi 207  df-an 396  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-dif 3906  df-nul 4288  df-sn 4583

This theorem is referenced by:  0nelop  5452  rnglidl0  21196  hausflim  23937  flimcf  23938  flimclslem  23940  cnpflf2  23956  cnpflf  23957  neipcfilu  24251  sltsbday  27925  zarclssn  34051  zar0ring  34056  elpaddat  40180  mnuprdlem1  44628  difmapsn  45570  ovnovollem1  47014  ovnovollem3  47016