Theorem snm 3743

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 3741	. 2 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ∈ {𝐴}

Syntax hints:  ∃wex 1506   ∈ wcel 2167  Vcvv 2763  {csn 3623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3629

This theorem is referenced by:  mss  4260  ssfilem  6945  diffitest  6957  djuexb  7119  exmidonfinlem  7272  exmidfodomrlemim  7280  cc2lem  7349