Theorem snm 3738

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 3736	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴}

Syntax hints:  ∃wex 1503   ∈ wcel 2164  Vcvv 2760  {csn 3618

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sn 3624

This theorem is referenced by:  mss  4255  ssfilem  6931  diffitest  6943  djuexb  7103  exmidonfinlem  7253  exmidfodomrlemim  7261  cc2lem  7326