Theorem snmg 3636

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

Assertion

Ref	Expression
snmg	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem snmg

Step	Hyp	Ref	Expression
1		snidg 3549	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		elex2 2697	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   → wi 4  ∃wex 1468   ∈ wcel 1480  {csn 3522

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sn 3528

This theorem is referenced by:  snm  3638  prmg  3639  exmidsssnc  4121  xpimasn  4982  1stconst  6111  2ndconst  6112