Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  predfrirr Unicode version

Theorem predfrirr 24269
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 4386 . . . . 5  |-  ( ( R  Fr  A  /\  X  e.  A )  ->  -.  X R X )
2 elpredg 24249 . . . . . . 7  |-  ( ( X  e.  A  /\  X  e.  A )  ->  ( X  e.  Pred ( R ,  A ,  X )  <->  X R X ) )
32anidms 626 . . . . . 6  |-  ( X  e.  A  ->  ( X  e.  Pred ( R ,  A ,  X
)  <->  X R X ) )
43notbid 285 . . . . 5  |-  ( X  e.  A  ->  ( -.  X  e.  Pred ( R ,  A ,  X )  <->  -.  X R X ) )
51, 4syl5ibr 212 . . . 4  |-  ( X  e.  A  ->  (
( R  Fr  A  /\  X  e.  A
)  ->  -.  X  e.  Pred ( R ,  A ,  X )
) )
65exp3a 425 . . 3  |-  ( X  e.  A  ->  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) ) )
76pm2.43b 46 . 2  |-  ( R  Fr  A  ->  ( X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) ) )
8 predel 24254 . . 3  |-  ( X  e.  Pred ( R ,  A ,  X )  ->  X  e.  A )
98con3i 127 . 2  |-  ( -.  X  e.  A  ->  -.  X  e.  Pred ( R ,  A ,  X ) )
107, 9pm2.61d1 151 1  |-  ( R  Fr  A  ->  -.  X  e.  Pred ( R ,  A ,  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   class class class wbr 4039    Fr wfr 4365   Predcpred 24238
This theorem is referenced by:  wfrlem14  24340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-fr 4368  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-pred 24239
  Copyright terms: Public domain W3C validator