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Theorem prelpwi 23187
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 3773 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
2 prex 4219 . . 3  |-  { A ,  B }  e.  _V
3 elpwg 3634 . . 3  |-  ( { A ,  B }  e.  _V  ->  ( { A ,  B }  e.  ~P C  <->  { A ,  B }  C_  C
) )
42, 3ax-mp 8 . 2  |-  ( { A ,  B }  e.  ~P C  <->  { A ,  B }  C_  C
)
51, 4sylibr 203 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1686   _Vcvv 2790    C_ wss 3154   ~Pcpw 3627   {cpr 3643
This theorem is referenced by:  measxun2  23540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-pw 3629  df-sn 3648  df-pr 3649
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