Theorem snexg 5429

Description: A singleton built on a set is a set. Special case of snex 5430 which does not require ax-nul 5305 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5430 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 4637	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		vsnex 5428	. . 3 ⊢ {𝑥} ∈ V
3	1, 2	eqeltrrdi 2840	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
4	3	vtocleg 3540	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  Vcvv 3472  {csn 4627

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3952  df-sn 4628  df-pr 4630

This theorem is referenced by:  snex  5430  selsALT  5438  snelpwg  5441  intidg  5456