Theorem snexgALT 5387

Description: Alternate proof of snexg 5386 based on vsnex 5381, which uses an instance of ax-sep 5243. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5385 and shorten proof. (Revised by BJ, 15-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem snexgALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4592	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		vsnex 5381	. . 3 ⊢ {𝑥} ∈ V
3	1, 2	eqeltrrdi 2846	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
4	3	vtocleg 3512	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-sn 4583  df-pr 4585

This theorem is referenced by: (None)