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Theorem ordirr 4267
 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 4266 . 2 ¬ 𝐴𝐴
21a1i 9 1 (Ord 𝐴 → ¬ 𝐴𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1393  Ord word 4099 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 4262 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 2311  df-v 2559  df-dif 2920  df-sn 3381 This theorem is referenced by:  nordeq  4268  ordn2lp  4269  orddisj  4270  onprc  4276  nlimsucg  4290  addnidpig  6434  frecfzennn  9203
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