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Theorem ordirr 4235
Description: Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. (Contributed by NM, 2-Jan-1994.)
Assertion
Ref Expression
ordirr (Ord 𝐴 → ¬ 𝐴𝐴)

Proof of Theorem ordirr
StepHypRef Expression
1 elirr 4234 . 2 ¬ 𝐴𝐴
21a1i 9 1 (Ord 𝐴 → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1393  Ord word 4070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4230
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-v 2556  df-dif 2917  df-sn 3378
This theorem is referenced by:  nordeq  4236  ordn2lp  4237  orddisj  4238  onprc  4244  nlimsucg  4258  addnidpig  6377  frecfzennn  9043
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