Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  predel Unicode version

Theorem predel 24254
Description: Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)
Assertion
Ref Expression
predel  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )

Proof of Theorem predel
StepHypRef Expression
1 elin 3371 . . 3  |-  ( Y  e.  ( A  i^i  ( `' R " { X } ) )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X } ) ) )
21simplbi 446 . 2  |-  ( Y  e.  ( A  i^i  ( `' R " { X } ) )  ->  Y  e.  A )
3 df-pred 24239 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
42, 3eleq2s 2388 1  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    i^i cin 3164   {csn 3653   `'ccnv 4704   "cima 4708   Predcpred 24238
This theorem is referenced by:  predpo  24255  predpoirr  24268  predfrirr  24269  dftrpred3g  24307
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-in 3172  df-pred 24239
  Copyright terms: Public domain W3C validator