ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prelpwi Unicode version

Theorem prelpwi 3978
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 3550 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
2 elex 2583 . . . 4  |-  ( A  e.  C  ->  A  e.  _V )
3 elex 2583 . . . 4  |-  ( B  e.  C  ->  B  e.  _V )
4 prexgOLD 3974 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
52, 3, 4syl2an 277 . . 3  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  _V )
6 elpwg 3395 . . 3  |-  ( { A ,  B }  e.  _V  ->  ( { A ,  B }  e.  ~P C  <->  { A ,  B }  C_  C
) )
75, 6syl 14 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( { A ,  B }  e.  ~P C 
<->  { A ,  B }  C_  C ) )
81, 7mpbird 160 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator