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Theorem snnz 3515
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 3513 . 2 (𝐴 ∈ V → {𝐴} ≠ ∅)
31, 2ax-mp 7 1 {𝐴} ≠ ∅
Colors of variables: wff set class
Syntax hints:  wcel 1409  wne 2220  Vcvv 2574  c0 3252  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2948  df-nul 3253  df-sn 3409
This theorem is referenced by:  0nep0  3946  1n0  6047
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