Theorem snm 3510

Description: The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)

Hypothesis

Ref	Expression
snnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snm	⊢ ∃𝑥 𝑥 ∈ {𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem snm

Step	Hyp	Ref	Expression
1		snnz.1	. 2 ⊢ 𝐴 ∈ V
2		snmg 3508	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	ax-mp 7	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴}

Syntax hints:  ∃wex 1421   ∈ wcel 1433  Vcvv 2601  {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by:  mss  3981  ssfilem  6360  diffitest  6371