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Theorem snnz 3723
Description: The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
snnz.1 𝐴 ∈ V
Assertion
Ref Expression
snnz {𝐴} ≠ ∅

Proof of Theorem snnz
StepHypRef Expression
1 snnz.1 . 2 𝐴 ∈ V
2 snnzg 3721 . 2 (𝐴 ∈ V → {𝐴} ≠ ∅)
31, 2ax-mp 5 1 {𝐴} ≠ ∅
Colors of variables: wff set class
Syntax hints:  wcel 2158  wne 2357  Vcvv 2749  c0 3434  {csn 3604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-nul 3435  df-sn 3610
This theorem is referenced by:  0nep0  4177  1n0  6447  ssfii  6987
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