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Theorem snelpwi 4134
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 3664 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 elex 2697 . . 3 (𝐴𝐵𝐴 ∈ V)
3 snexg 4108 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
4 elpwg 3518 . . 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 1480  Vcvv 2686  wss 3071  𝒫 cpw 3510  {csn 3527
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533
This theorem is referenced by:  unipw  4139  infpwfidom  7054  txdis  12456  txdis1cn  12457
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