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Theorem snelpwi 3939
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 (A B → {A} 𝒫 B)

Proof of Theorem snelpwi
StepHypRef Expression
1 snssi 3499 . 2 (A B → {A} ⊆ B)
2 elex 2560 . . 3 (A BA V)
3 snexgOLD 3926 . . 3 (A V → {A} V)
4 elpwg 3359 . . 3 ({A} V → ({A} 𝒫 B ↔ {A} ⊆ B))
52, 3, 43syl 17 . 2 (A B → ({A} 𝒫 B ↔ {A} ⊆ B))
61, 5mpbird 156 1 (A B → {A} 𝒫 B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  Vcvv 2551  wss 2911  𝒫 cpw 3351  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by:  unipw  3944
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