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Theorem prelpwi 4353
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 3898 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
2 prex 4348 . . 3  |-  { A ,  B }  e.  _V
32elpw 3749 . 2  |-  ( { A ,  B }  e.  ~P C  <->  { A ,  B }  C_  C
)
41, 3sylibr 204 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717    C_ wss 3264   ~Pcpw 3743   {cpr 3759
This theorem is referenced by:  inelfi  7359  isdrs2  14324  usgra1  21261  usgraexmpl  21287  cusgraexi  21344  cusgrafilem2  21356  unelsiga  24314  measxun2  24359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-pw 3745  df-sn 3764  df-pr 3765
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