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Theorem ordirr 5658
 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 → ¬ 𝐴𝐴)

Proof of Theorem ordirr
StepHypRef Expression
1 ordfr 5655 . 2 (Ord 𝐴 → E Fr 𝐴)
2 efrirr 5019 . 2 ( E Fr 𝐴 → ¬ 𝐴𝐴)
31, 2syl 17 1 (Ord 𝐴 → ¬ 𝐴𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977   E cep 4947   Fr wfr 4994  Ord word 5639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997  df-we 4999  df-ord 5643 This theorem is referenced by:  nordeq  5659  ordn2lp  5660  ordtri3or  5672  ordtri1  5673  ordtri3  5676  ordtri3OLD  5677  orddisj  5679  ordunidif  5690  ordnbtwn  5733  ordnbtwnOLD  5734  onirri  5751  onssneli  5754  onprc  6876  nlimsucg  6934  nnlim  6970  limom  6972  smo11  7348  smoord  7349  tfrlem13  7373  omopth2  7551  limensuci  8021  infensuc  8023  ordtypelem9  8314  cantnfp1lem3  8460  cantnfp1  8461  oemapvali  8464  tskwe  8659  dif1card  8716  pm110.643ALT  8883  pwsdompw  8909  cflim2  8968  fin23lem24  9027  fin23lem26  9030  axdc3lem4  9158  ttukeylem7  9220  canthp1lem2  9354  inar1  9476  gruina  9519  grur1  9521  addnidpi  9602  fzennn  12629  hashp1i  13052  soseq  30995  noseponlem  31065  sucneqond  32389
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