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Theorem snelpw 4840
 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 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpw
StepHypRef Expression
1 snelpw.1 . . 3 𝐴 ∈ V
21snss 4259 . 2 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
3 snex 4835 . . 3 {𝐴} ∈ V
43elpw 4114 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
52, 4bitr4i 266 1 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128 This theorem is referenced by:  dis2ndc  21073  dislly  21110  upgredg  25811
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