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 17	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 2932  ∅c0 4285  {csn 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-dif 3904  df-nul 4286  df-sn 4581

This theorem is referenced by:  0nelop  5444  rnglidl0  21184  hausflim  23925  flimcf  23926  flimclslem  23928  cnpflf2  23944  cnpflf  23945  neipcfilu  24239  sltsbday  27913  zarclssn  34030  zar0ring  34035  elpaddat  40064  mnuprdlem1  44513  difmapsn  45456  ovnovollem1  46900  ovnovollem3  46902