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Theorem snelpw 5102
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 4502 . 2 (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)
3 snex 5097 . . 3 {𝐴} ∈ V
43elpw 4353 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
52, 4bitr4i 270 1 (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2157  Vcvv 3383  wss 3767  𝒫 cpw 4347  {csn 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-pw 4349  df-sn 4367  df-pr 4369
This theorem is referenced by:  dis2ndc  21588  dislly  21625
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