Theorem snn0d 4732

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 4731	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	1, 2	syl 18	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955  ∅c0 4277  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-dif 4046  df-nul 4278  df-sn 4585

This theorem is used by:  0nelop  5468  rnglidl0  21518  hausflim  24309  flimcf  24310  flimclslem  24312  cnpflf2  24328  cnpflf  24329  neipcfilu  24623  sltsbday  28312  zarclssn  34555  zar0ring  34560  elpaddat  40998  mnuprdlem1  45393  difmapsn  46361  ovnovollem1  47802  ovnovollem3  47804