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Theorem prelpwi 4637
Description: A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
Assertion
Ref Expression
prelpwi  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C
)

Proof of Theorem prelpwi
StepHypRef Expression
1 prssi 4127 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
2 prex 4632 . . 3  |-  { A ,  B }  e.  _V
32elpw 3960 . 2  |-  ( { A ,  B }  e.  ~P C  <->  { A ,  B }  C_  C
)
41, 3sylibr 212 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842    C_ wss 3413   ~Pcpw 3954   {cpr 3973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-pw 3956  df-sn 3972  df-pr 3974
This theorem is referenced by:  inelfi  7911  isdrs2  15890  usgra1  24777  usgraexmpl  24805  cusgraexi  24872  cusgrafilem2  24884  unelsiga  28568  unelldsys  28592  measxun2  28644  lincvalpr  38511  ldepspr  38566  zlmodzxzldeplem3  38595  zlmodzxzldep  38597  ldepsnlinc  38601
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