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

Theorem elpwi 3573
Description: Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
elpwi  |-  ( A  e.  ~P B  ->  A  C_  B )

Proof of Theorem elpwi
StepHypRef Expression
1 elpwg 3572 . 2  |-  ( A  e.  ~P B  -> 
( A  e.  ~P B 
<->  A  C_  B )
)
21ibi 175 1  |-  ( A  e.  ~P B  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141    C_ wss 3121   ~Pcpw 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566
This theorem is referenced by:  elpwid  3575  elelpwi  3576  elpw2g  4140  eldifpw  4460  iunpw  4463  f1opw2  6052  pw1dc1  6887  fi0  6948  pw1on  7190  cnntr  12940
  Copyright terms: Public domain W3C validator