Theorem snexgALT 5414

Description: Alternate proof of snexg 5413 based on vsnex 5408, which uses an instance of ax-sep 5259. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5412 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 4601	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		vsnex 5408	. . 3 ⊢ {𝑥} ∈ V
3	1, 2	eqeltrrdi 2874	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
4	3	vtocleg 3523	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-sn 4592  df-pr 4594

This theorem is used by: (None)