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

Theorem elpredim 24176
Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
Hypothesis
Ref Expression
elpredim.1  |-  X  e. 
_V
Assertion
Ref Expression
elpredim  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X )

Proof of Theorem elpredim
StepHypRef Expression
1 df-pred 24168 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
21elin2 3359 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) ) )
3 elpredim.1 . . . . . 6  |-  X  e. 
_V
4 elimasng 5039 . . . . . . 7  |-  ( ( X  e.  _V  /\  Y  e.  ( `' R " { X }
) )  ->  ( Y  e.  ( `' R " { X }
)  <->  <. X ,  Y >.  e.  `' R ) )
5 opelcnvg 4861 . . . . . . 7  |-  ( ( X  e.  _V  /\  Y  e.  ( `' R " { X }
) )  ->  ( <. X ,  Y >.  e.  `' R  <->  <. Y ,  X >.  e.  R ) )
64, 5bitrd 244 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  ( `' R " { X }
) )  ->  ( Y  e.  ( `' R " { X }
)  <->  <. Y ,  X >.  e.  R ) )
73, 6mpan 651 . . . . 5  |-  ( Y  e.  ( `' R " { X } )  ->  ( Y  e.  ( `' R " { X } )  <->  <. Y ,  X >.  e.  R ) )
87ibi 232 . . . 4  |-  ( Y  e.  ( `' R " { X } )  ->  <. Y ,  X >.  e.  R )
9 df-br 4024 . . . 4  |-  ( Y R X  <->  <. Y ,  X >.  e.  R )
108, 9sylibr 203 . . 3  |-  ( Y  e.  ( `' R " { X } )  ->  Y R X )
1110adantl 452 . 2  |-  ( ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) )  ->  Y R X )
122, 11sylbi 187 1  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643   class class class wbr 4023   `'ccnv 4688   "cima 4692   Predcpred 24167
This theorem is referenced by:  predbrg  24186  preddowncl  24196  trpredrec  24241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
  Copyright terms: Public domain W3C validator