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Theorem snelpwi 4050
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 3589 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 elex 2633 . . 3 (𝐴𝐵𝐴 ∈ V)
3 snexg 4027 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
4 elpwg 3443 . . 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 1439  Vcvv 2622  wss 3002  𝒫 cpw 3435  {csn 3452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458
This theorem is referenced by:  unipw  4055  infpwfidom  6887
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