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Theorem prsspw 3565
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 3549 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C )
41elpw 3396 . . 3  |-  ( A  e.  ~P C  <->  A  C_  C
)
52elpw 3396 . . 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 1434   _Vcvv 2602    C_ wss 2974   ~Pcpw 3390   {cpr 3407
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413
This theorem is referenced by: (None)
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