Theorem snm 3703

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 3701	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴}

Syntax hints:  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3589

This theorem is referenced by:  mss  4211  ssfilem  6853  diffitest  6865  djuexb  7021  exmidonfinlem  7170  exmidfodomrlemim  7178  cc2lem  7228