Theorem snelpwg 5453

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 5312. (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 4788	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
2		snexg 5441	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		elpwg 4608	. . 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 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634

This theorem is referenced by:  snelpwi  5454  snelpw  5456