Theorem snn0d 4711

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 4710	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-dif 3890  df-nul 4257  df-sn 4562

This theorem is referenced by:  0nelop  5410  hausflim  23132  flimcf  23133  flimclslem  23135  cnpflf2  23151  cnpflf  23152  neipcfilu  23448  zarclssn  31823  zar0ring  31828  elpaddat  37818  mnuprdlem1  41890  difmapsn  42752  ovnovollem1  44194  ovnovollem3  44196