Theorem snexg 5387

Description: A singleton built on a set is a set. Special case of snex 5386 which is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5386 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 5386	. 2 ⊢ {𝐴} ∈ V
2	1	a1i 11	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2132  Vcvv 3444  {csn 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-v 3446  df-un 3900  df-sn 4573  df-pr 4575

This theorem is referenced by:  snexOLD  5389  selsALT  5398  snelpwg  5400  intidg  5414  oncutlt  28323  elreno2  28554