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Theorem snnz 3606
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 3604 . 2 (𝐴 ∈ V → {𝐴} ≠ ∅)
31, 2ax-mp 7 1 {𝐴} ≠ ∅
Colors of variables: wff set class
Syntax hints:  wcel 1461  wne 2280  Vcvv 2655  c0 3327  {csn 3491
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-v 2657  df-dif 3037  df-nul 3328  df-sn 3497
This theorem is referenced by:  0nep0  4047  1n0  6281  ssfii  6812
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