Theorem snelpwg 5410

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 5256. (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 5397	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		elpwg 4558	. . 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 284	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-pw 4557  df-sn 4583  df-pr 4585

This theorem is referenced by:  snelpwi  5411  snelpw  5412