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Theorem snn0d 4674
 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 4673 . 2 (𝐴𝑉 → {𝐴} ≠ ∅)
31, 2syl 17 1 (𝜑 → {𝐴} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   ≠ wne 2990  ∅c0 4246  {csn 4528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-dif 3887  df-nul 4247  df-sn 4529 This theorem is referenced by:  zar0ring  31231  difmapsn  41828  ovnovollem1  43282
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