Theorem snn0d 4736

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 4735	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	1, 2	syl 18	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2956  ∅c0 4279  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-dif 3902  df-nul 4280  df-sn 4585

This theorem is used by:  0nelop  5468  rnglidl0  21509  hausflim  24300  flimcf  24301  flimclslem  24303  cnpflf2  24319  cnpflf  24320  neipcfilu  24614  sltsbday  28303  zarclssn  34505  zar0ring  34510  elpaddat  40861  mnuprdlem1  45255  difmapsn  46224  ovnovollem1  47665  ovnovollem3  47667