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

Theorem prsspwg 3514
Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
Assertion
Ref Expression
prsspwg  V  W  { ,  }  C_  ~P C  C_  C  C_  C

Proof of Theorem prsspwg
StepHypRef Expression
1 prssg 3512 . 2  V  W  ~P C  ~P C  { ,  }  C_  ~P C
2 elpwg 3359 . . 3  V  ~P C  C_  C
3 elpwg 3359 . . 3  W  ~P C  C_  C
42, 3bi2anan9 538 . 2  V  W  ~P C  ~P C 
C_  C  C_  C
51, 4bitr3d 179 1  V  W  { ,  }  C_  ~P C  C_  C  C_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wcel 1390    C_ wss 2911   ~Pcpw 3351   {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator