Theorem snexg 5426

Description: A singleton built on a set is a set. Special case of snex 5427 which does not require ax-nul 5302 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5427 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 4634	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		vsnex 5425	. . 3 ⊢ {𝑥} ∈ V
3	1, 2	eqeltrrdi 2843	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
4	3	vtocleg 3544	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3951  df-sn 4625  df-pr 4627

This theorem is referenced by:  snex  5427  selsALT  5435  snelpwg  5438  intidg  5453