Theorem snelpw 5355

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Hypothesis

Ref	Expression
snelpw.1	⊢ 𝐴 ∈ V

Assertion

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

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. . 3 ⊢ 𝐴 ∈ V
2	1	snss 4716	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
3		snex 5349	. . 3 ⊢ {𝐴} ∈ V
4	3	elpw 4534	. 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
5	2, 4	bitr4i 277	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 205   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561

This theorem is referenced by:  pwfir  8921  dis2ndc  22519  dislly  22556