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Theorem predfrirr 25465
Description: Given a well-founded relationship,  X is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
Assertion
Ref Expression
predfrirr  |-  ( R  Fr  A  ->  -.  X  e.  Pred ( R ,  A ,  X
) )

Proof of Theorem predfrirr
StepHypRef Expression
1 frirr 4551 . . . . 5  |-  ( ( R  Fr  A  /\  X  e.  A )  ->  -.  X R X )
2 elpredg 25445 . . . . . . 7  |-  ( ( X  e.  A  /\  X  e.  A )  ->  ( X  e.  Pred ( R ,  A ,  X )  <->  X R X ) )
32anidms 627 . . . . . 6  |-  ( X  e.  A  ->  ( X  e.  Pred ( R ,  A ,  X
)  <->  X R X ) )
43notbid 286 . . . . 5  |-  ( X  e.  A  ->  ( -.  X  e.  Pred ( R ,  A ,  X )  <->  -.  X R X ) )
51, 4syl5ibr 213 . . . 4  |-  ( X  e.  A  ->  (
( R  Fr  A  /\  X  e.  A
)  ->  -.  X  e.  Pred ( R ,  A ,  X )
) )
65exp3a 426 . . 3  |-  ( X  e.  A  ->  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) ) )
76pm2.43b 48 . 2  |-  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) )
8 predel 25450 . . 3  |-  ( X  e.  Pred ( R ,  A ,  X )  ->  X  e.  A )
98con3i 129 . 2  |-  ( -.  X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) )
107, 9pm2.61d1 153 1  |-  ( R  Fr  A  ->  -.  X  e.  Pred ( R ,  A ,  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   class class class wbr 4204    Fr wfr 4530   Predcpred 25430
This theorem is referenced by:  wfrlem14  25543
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-fr 4533  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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