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Theorem snnzg 3459
Description: The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
snnzg (A 𝑉 → {A} ≠ ∅)

Proof of Theorem snnzg
StepHypRef Expression
1 snidg 3375 . 2 (A 𝑉A {A})
2 ne0i 3207 . 2 (A {A} → {A} ≠ ∅)
31, 2syl 14 1 (A 𝑉 → {A} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  wne 2186  c0 3201  {csn 3350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-nul 3202  df-sn 3356
This theorem is referenced by:  snnz  3461  0nelop  3959
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