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Theorem ordirr 4427
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4422. If in the definition of ordinals df-iord 4258, we also required that membership be well-founded on any ordinal (see df-frind 4224), then we could prove ordirr 4427 without ax-setind 4422. (Contributed by NM, 2-Jan-1994.)
Assertion
Ref Expression
ordirr (Ord 𝐴 → ¬ 𝐴𝐴)

Proof of Theorem ordirr
StepHypRef Expression
1 elirr 4426 . 2 ¬ 𝐴𝐴
21a1i 9 1 (Ord 𝐴 → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1465  Ord word 4254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-v 2662  df-dif 3043  df-sn 3503
This theorem is referenced by:  onirri  4428  nordeq  4429  ordn2lp  4430  orddisj  4431  onprc  4437  nlimsucg  4451  tfr1onlemsucfn  6205  tfr1onlemsucaccv  6206  tfrcllemsucfn  6218  tfrcllemsucaccv  6219  nntr2  6367  unsnfi  6775  addnidpig  7112  frecfzennn  10167  hashinfom  10492  hashennn  10494  hashp1i  10524  ennnfonelemg  11843  ctinfom  11868  nninfalllemn  13129
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