Theorem snelpw 5397

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 5395	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3429  𝒫 cpw 4541  {csn 4567

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-pw 4543  df-sn 4568  df-pr 4570

This theorem is referenced by:  pwfir  9227  dis2ndc  23425  dislly  23462  n0cut  28326  n0on  28328  mh-infprim2bi  36729