MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prsspw Unicode version

Theorem prsspw 3801
Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
prsspw.1  |-  A  e. 
_V
prsspw.2  |-  B  e. 
_V
Assertion
Ref Expression
prsspw  |-  ( { A ,  B }  C_ 
~P C  <->  ( A  C_  C  /\  B  C_  C ) )

Proof of Theorem prsspw
StepHypRef Expression
1 prsspw.1 . . 3  |-  A  e. 
_V
2 prsspw.2 . . 3  |-  B  e. 
_V
31, 2prss 3785 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C )
41elpw 3644 . . 3  |-  ( A  e.  ~P C  <->  A  C_  C
)
52elpw 3644 . . 3  |-  ( B  e.  ~P C  <->  B  C_  C
)
64, 5anbi12i 678 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) )
73, 6bitr3i 242 1  |-  ( { A ,  B }  C_ 
~P C  <->  ( A  C_  C  /\  B  C_  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   {cpr 3654
This theorem is referenced by:  altxpsspw  24583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-pw 3640  df-sn 3659  df-pr 3660
  Copyright terms: Public domain W3C validator