Theorem snelpw 5445

Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.)

Hypothesis

Ref	Expression
snelpw.ex	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snelpw	⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.ex	. 2 ⊢ 𝐴 ∈ V
2		snelpwg 5442	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 205   ∈ wcel 2107  Vcvv 3475  𝒫 cpw 4602  {csn 4628

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 5299  ax-pr 5427

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 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-pr 4631

This theorem is referenced by:  pwfir  9173  dis2ndc  22956  dislly  22993