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Theorem elpred 25194
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
Hypothesis
Ref Expression
elpred.1  |-  Y  e. 
_V
Assertion
Ref Expression
elpred  |-  ( X  e.  D  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  ( Y  e.  A  /\  Y R X ) ) )

Proof of Theorem elpred
StepHypRef Expression
1 df-pred 25185 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
21elin2 3467 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) ) )
3 elpred.1 . . . 4  |-  Y  e. 
_V
43eliniseg 5166 . . 3  |-  ( X  e.  D  ->  ( Y  e.  ( `' R " { X }
)  <->  Y R X ) )
54anbi2d 685 . 2  |-  ( X  e.  D  ->  (
( Y  e.  A  /\  Y  e.  ( `' R " { X } ) )  <->  ( Y  e.  A  /\  Y R X ) ) )
62, 5syl5bb 249 1  |-  ( X  e.  D  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  ( Y  e.  A  /\  Y R X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   _Vcvv 2892   {csn 3750   class class class wbr 4146   `'ccnv 4810   "cima 4814   Predcpred 25184
This theorem is referenced by:  predreseq  25196  predpo  25201  setlikespec  25204  preddowncl  25213  preduz  25217  predfz  25220  wfrlem10  25282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-pred 25185
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