Theorem snelpwg 5398

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.) Put in closed form and avoid ax-nul 5253. (Revised by BJ, 17-Jan-2025.)

Assertion

Ref	Expression
snelpwg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))

Proof of Theorem snelpwg

Step	Hyp	Ref	Expression
1		snssg 4742	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
2		snexg 5386	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		elpwg 4559	. . 3 ⊢ ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
5	1, 4	bitr4d 282	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-pw 4558  df-sn 4583  df-pr 4585

This theorem is referenced by:  snelpwi  5399  snelpw  5400