ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snnzg GIF version

Theorem snnzg 3587
Description: The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
snnzg (𝐴𝑉 → {𝐴} ≠ ∅)

Proof of Theorem snnzg
StepHypRef Expression
1 snidg 3501 . 2 (𝐴𝑉𝐴 ∈ {𝐴})
2 ne0i 3316 . 2 (𝐴 ∈ {𝐴} → {𝐴} ≠ ∅)
31, 2syl 14 1 (𝐴𝑉 → {𝐴} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1448  wne 2267  c0 3310  {csn 3474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-v 2643  df-dif 3023  df-nul 3311  df-sn 3480
This theorem is referenced by:  snnz  3589  0nelop  4108
  Copyright terms: Public domain W3C validator