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

Theorem predfrirr 27779
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 4781 . . . . 5  |-  ( ( R  Fr  A  /\  X  e.  A )  ->  -.  X R X )
2 elpredg 27759 . . . . . . 7  |-  ( ( X  e.  A  /\  X  e.  A )  ->  ( X  e.  Pred ( R ,  A ,  X )  <->  X R X ) )
32anidms 645 . . . . . 6  |-  ( X  e.  A  ->  ( X  e.  Pred ( R ,  A ,  X
)  <->  X R X ) )
43notbid 294 . . . . 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 436 . . 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 27764 . . 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 369    e. wcel 1757   class class class wbr 4376    Fr wfr 4760   Predcpred 27744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-fr 4763  df-xp 4930  df-cnv 4932  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-pred 27745
This theorem is referenced by:  wfrlem14  27857
  Copyright terms: Public domain W3C validator