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Theorem predfrirr 5416
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 4816 . . . . 5  |-  ( ( R  Fr  A  /\  X  e.  A )  ->  -.  X R X )
2 elpredg 5401 . . . . . . 7  |-  ( ( X  e.  A  /\  X  e.  A )  ->  ( X  e.  Pred ( R ,  A ,  X )  <->  X R X ) )
32anidms 657 . . . . . 6  |-  ( X  e.  A  ->  ( X  e.  Pred ( R ,  A ,  X
)  <->  X R X ) )
43notbid 301 . . . . 5  |-  ( X  e.  A  ->  ( -.  X  e.  Pred ( R ,  A ,  X )  <->  -.  X R X ) )
51, 4syl5ibr 229 . . . 4  |-  ( X  e.  A  ->  (
( R  Fr  A  /\  X  e.  A
)  ->  -.  X  e.  Pred ( R ,  A ,  X )
) )
65expd 443 . . 3  |-  ( X  e.  A  ->  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) ) )
76pm2.43b 51 . 2  |-  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) )
8 predel 5404 . . 3  |-  ( X  e.  Pred ( R ,  A ,  X )  ->  X  e.  A )
98con3i 142 . 2  |-  ( -.  X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) )
107, 9pm2.61d1 164 1  |-  ( R  Fr  A  ->  -.  X  e.  Pred ( R ,  A ,  X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    e. wcel 1904   class class class wbr 4395    Fr wfr 4795   Predcpred 5386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-fr 4798  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387
This theorem is referenced by:  wfrlem14  7067
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