Theorem ordirr 4356

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 4351. If in the definition of ordinals df-iord 4191, we also required that membership be well-founded on any ordinal (see df-frind 4157), then we could prove ordirr 4356 without ax-setind 4351. (Contributed by NM, 2-Jan-1994.)

Assertion

Ref	Expression
ordirr	⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4355	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1438  Ord word 4187

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4351

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-sn 3450

This theorem is referenced by:  onirri  4357  nordeq  4358  ordn2lp  4359  orddisj  4360  onprc  4366  nlimsucg  4380  tfr1onlemsucfn  6097  tfr1onlemsucaccv  6098  tfrcllemsucfn  6110  tfrcllemsucaccv  6111  unsnfi  6619  addnidpig  6885  frecfzennn  9821  hashinfom  10174  hashennn  10176  hashp1i  10206  nninfalllemn  11781