Theorem snexg 5371

Description: A singleton built on a set is a set. Special case of snex 5370 which is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5370 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 5370	. 2 ⊢ {𝐴} ∈ V
2	1	a1i 11	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3427  {csn 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3429  df-un 3890  df-sn 4558  df-pr 4560

This theorem is referenced by:  snexOLD  5373  selsALT  5382  snelpwg  5384  intidg  5398  oncutlt  28244  elreno2  28475