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

Theorem prsspw 3609
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 3593 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C )
41elpw 3435 . . 3  |-  ( A  e.  ~P C  <->  A  C_  C
)
52elpw 3435 . . 3  |-  ( B  e.  ~P C  <->  B  C_  C
)
64, 5anbi12i 448 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) )
73, 6bitr3i 184 1  |-  ( { A ,  B }  C_ 
~P C  <->  ( A  C_  C  /\  B  C_  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1438   _Vcvv 2619    C_ wss 2999   ~Pcpw 3429   {cpr 3447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator