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Theorem elpred 25444
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 25431 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
21elin2 3523 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) ) )
3 elpred.1 . . . 4  |-  Y  e. 
_V
43eliniseg 5225 . . 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 1725   _Vcvv 2948   {csn 3806   class class class wbr 4204   `'ccnv 4869   "cima 4873   Predcpred 25430
This theorem is referenced by:  predreseq  25446  predpo  25451  setlikespec  25454  preddowncl  25463  preduz  25467  predfz  25470  wfrlem10  25539  wzel  25567  wsuclem  25568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-pred 25431
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