Theorem snexg 5397

Description: A singleton built on a set is a set. Special case of snex 5396 which is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5396 and shorten proof. (Revised by BJ, 15-Jan-2025.) (Proof shortened by GG, 6-Mar-2026.)

Assertion

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

Proof of Theorem snexg

Step	Hyp	Ref	Expression
1		snex 5396	. 2 ⊢ {𝐴} ∈ V
2	1	a1i 11	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3450  {csn 4583

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3903  df-sn 4584  df-pr 4586

This theorem is used by:  snexOLD  5399  selsALT  5408  snelpwg  5410  intidg  5424  oncutlt  28584  elreno2  28815  negprop  38563