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Theorem snelpw 3940
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  _V
Assertion
Ref Expression
snelpw  { }  ~P

Proof of Theorem snelpw
StepHypRef Expression
1 snelpw.1 . . 3  _V
21snss 3485 . 2  { }  C_
3 snexgOLD 3926 . . . 4  _V  { }  _V
41, 3ax-mp 7 . . 3  { }  _V
54elpw 3357 . 2  { }  ~P  { }  C_
62, 5bitr4i 176 1  { }  ~P
Colors of variables: wff set class
Syntax hints:   wb 98   wcel 1390   _Vcvv 2551    C_ wss 2911   ~Pcpw 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-bndl 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: (None)
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