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Theorem predfrirr 29521
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 4845 . . . . 5  |-  ( ( R  Fr  A  /\  X  e.  A )  ->  -.  X R X )
2 elpredg 29501 . . . . . . 7  |-  ( ( X  e.  A  /\  X  e.  A )  ->  ( X  e.  Pred ( R ,  A ,  X )  <->  X R X ) )
32anidms 643 . . . . . 6  |-  ( X  e.  A  ->  ( X  e.  Pred ( R ,  A ,  X
)  <->  X R X ) )
43notbid 292 . . . . 5  |-  ( X  e.  A  ->  ( -.  X  e.  Pred ( R ,  A ,  X )  <->  -.  X R X ) )
51, 4syl5ibr 221 . . . 4  |-  ( X  e.  A  ->  (
( R  Fr  A  /\  X  e.  A
)  ->  -.  X  e.  Pred ( R ,  A ,  X )
) )
65expd 434 . . 3  |-  ( X  e.  A  ->  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) ) )
76pm2.43b 50 . 2  |-  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) )
8 predel 29506 . . 3  |-  ( X  e.  Pred ( R ,  A ,  X )  ->  X  e.  A )
98con3i 135 . 2  |-  ( -.  X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) )
107, 9pm2.61d1 159 1  |-  ( R  Fr  A  ->  -.  X  e.  Pred ( R ,  A ,  X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823   class class class wbr 4439    Fr wfr 4824   Predcpred 29486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-fr 4827  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-pred 29487
This theorem is referenced by:  wfrlem14  29599
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