Theorem snexg 5399

Description: A singleton built on a set is a set. Special case of snex 5398 which is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5398 and shorten proof. (Revised by BJ, 15-Jan-2025.) (Proof shortened by GG, 6-Mar-2026.)

Assertion

Ref	Expression
snexg	⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Proof of Theorem snexg

Step	Hyp	Ref	Expression
1		snex 5398	. 2 ⊢ {𝐴} ∈ V
2	1	a1i 11	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2144  Vcvv 3456  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-un 3911  df-sn 4585  df-pr 4587

This theorem is referenced by:  snexOLD  5401  selsALT  5410  snelpwg  5412  intidg  5426  oncutlt  28359  elreno2  28590