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Theorem snn0d 38284
 Description: The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
snn0d.1 (𝜑𝐴𝑉)
Assertion
Ref Expression
snn0d (𝜑 → {𝐴} ≠ ∅)

Proof of Theorem snn0d
StepHypRef Expression
1 snn0d.1 . 2 (𝜑𝐴𝑉)
2 snnzg 4251 . 2 (𝐴𝑉 → {𝐴} ≠ ∅)
31, 2syl 17 1 (𝜑 → {𝐴} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-nul 3875  df-sn 4126 This theorem is referenced by:  difmapsn  38399  ovnovollem1  39546
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