Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  predfrirr Structured version   Visualization version   Unicode version

Theorem predfrirr 5416
 Description: Given a well-founded relationship, is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
Assertion
Ref Expression
predfrirr

Proof of Theorem predfrirr
StepHypRef Expression
1 frirr 4816 . . . . 5
2 elpredg 5401 . . . . . . 7
32anidms 657 . . . . . 6
43notbid 301 . . . . 5
51, 4syl5ibr 229 . . . 4
65expd 443 . . 3
76pm2.43b 51 . 2
8 predel 5404 . . 3
98con3i 142 . 2
107, 9pm2.61d1 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wa 376   wcel 1904   class class class wbr 4395   wfr 4795  cpred 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
 Copyright terms: Public domain W3C validator