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Theorem snelpwi 4104
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
Assertion
Ref Expression
snelpwi (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwi
StepHypRef Expression
1 snssi 3634 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 elex 2671 . . 3 (𝐴𝐵𝐴 ∈ V)
3 snexg 4078 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
4 elpwg 3488 . . 3 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
52, 3, 43syl 17 . 2 (𝐴𝐵 → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
61, 5mpbird 166 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1465  Vcvv 2660  wss 3041  𝒫 cpw 3480  {csn 3497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503
This theorem is referenced by:  unipw  4109  infpwfidom  7022  txdis  12373  txdis1cn  12374
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