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Theorem ordirr 4426
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.)
Assertion
Ref Expression
ordirr  |-  ( Ord 
A  ->  -.  A  e.  A )

Proof of Theorem ordirr
StepHypRef Expression
1 ordfr 4423 . 2  |-  ( Ord 
A  ->  _E  Fr  A )
2 efrirr 4390 . 2  |-  (  _E  Fr  A  ->  -.  A  e.  A )
31, 2syl 15 1  |-  ( Ord 
A  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1696    _E cep 4319    Fr wfr 4365   Ord word 4407
This theorem is referenced by:  nordeq  4427  ordn2lp  4428  ordtri3or  4440  ordtri1  4441  ordtri3  4444  orddisj  4446  ordunidif  4456  ordnbtwn  4499  onirri  4515  onssneli  4518  onprc  4592  nlimsucg  4649  nnlim  4685  limom  4687  smo11  6397  smoord  6398  tfrlem13  6422  omopth2  6598  limensuci  7053  infensuc  7055  ordtypelem9  7257  cantnfp1lem3  7398  cantnfp1  7399  oemapvali  7402  wfelirr  7513  tskwe  7599  dif1card  7654  pm110.643ALT  7820  pwsdompw  7846  cflim2  7905  fin23lem24  7964  fin23lem26  7967  axdc3lem4  8095  ttukeylem7  8158  canthp1lem2  8291  inar1  8413  gruina  8456  grur1  8458  addnidpi  8541  fzennn  11046  hashp1i  11385  soseq  24325  hfninf  24888
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-eprel 4321  df-fr 4368  df-we 4370  df-ord 4411
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