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Theorem prelpwi 4539
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 4029 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
2 prex 4534 . . 3  |-  { A ,  B }  e.  _V
32elpw 3866 . 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 1756    C_ wss 3328   ~Pcpw 3860   {cpr 3879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-pw 3862  df-sn 3878  df-pr 3880
This theorem is referenced by:  inelfi  7668  isdrs2  15109  usgra1  23292  usgraexmpl  23319  cusgraexi  23376  cusgrafilem2  23388  unelsiga  26577  measxun2  26624  lincvalpr  30952  ldepspr  31007  zlmodzxzldeplem3  31044  zlmodzxzldep  31046  ldepsnlinc  31050
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