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Theorem prelpwi 4700
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 4189 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
2 prex 4695 . . 3  |-  { A ,  B }  e.  _V
32elpw 4022 . 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 369    e. wcel 1767    C_ wss 3481   ~Pcpw 4016   {cpr 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034  df-pr 4036
This theorem is referenced by:  inelfi  7890  isdrs2  15442  usgra1  24205  usgraexmpl  24233  cusgraexi  24300  cusgrafilem2  24312  unelsiga  27966  measxun2  28013  lincvalpr  32506  ldepspr  32561  zlmodzxzldeplem3  32590  zlmodzxzldep  32592  ldepsnlinc  32596
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