Theorem snexg 5405

Description: A singleton built on a set is a set. Special case of snex 5406 which does not require ax-nul 5276 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5406 and shorten proof. (Revised by BJ, 15-Jan-2025.)

Assertion

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

Proof of Theorem snexg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4611	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		vsnex 5404	. . 3 ⊢ {𝑥} ∈ V
3	1, 2	eqeltrrdi 2843	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
4	3	vtocleg 3532	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459  {csn 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-sn 4602  df-pr 4604

This theorem is referenced by:  snex  5406  selsALT  5414  snelpwg  5417  intidg  5432  onscutlt  28217