Theorem snmg 3580

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 3493	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		elex2 2649	. 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝑥 ∈ {𝐴})
3	1, 2	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})

Syntax hints:   → wi 4  ∃wex 1433   ∈ wcel 1445  {csn 3466

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-sn 3472

This theorem is referenced by:  snm  3582  prmg  3583  xpimasn  4913  1stconst  6024  2ndconst  6025