Theorem snexgALT 5397

Description: Alternate proof of snexg 5396 based on vsnex 5391, which uses an instance of ax-sep 5245. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5395 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 4591	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2		vsnex 5391	. . 3 ⊢ {𝑥} ∈ V
3	1, 2	eqeltrrdi 2870	. 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V)
4	3	vtocleg 3520	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {csn 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4582  df-pr 4584

This theorem is referenced by: (None)