Theorem snmg 3701

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 3612	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		elex2 2746	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   → wi 4  ∃wex 1485   ∈ wcel 2141  {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:  snm  3703  prmg  3704  exmidsssnc  4189  xpimasn  5059  1stconst  6200  2ndconst  6201