Theorem snm 3714

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 3712	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴}

Syntax hints:  ∃wex 1492   ∈ wcel 2148  Vcvv 2739  {csn 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sn 3600

This theorem is referenced by:  mss  4228  ssfilem  6877  diffitest  6889  djuexb  7045  exmidonfinlem  7194  exmidfodomrlemim  7202  cc2lem  7267