Theorem ordirr 4465

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 4460. If in the definition of ordinals df-iord 4296, we also required that membership be well-founded on any ordinal (see df-frind 4262), then we could prove ordirr 4465 without ax-setind 4460. (Contributed by NM, 2-Jan-1994.)

Assertion

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

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4464	. 2 ⊢ ¬ 𝐴 ∈ 𝐴
2	1	a1i 9	1 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1481  Ord word 4292

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-v 2691  df-dif 3078  df-sn 3538

This theorem is referenced by:  onirri  4466  nordeq  4467  ordn2lp  4468  orddisj  4469  onprc  4475  nlimsucg  4489  tfr1onlemsucfn  6245  tfr1onlemsucaccv  6246  tfrcllemsucfn  6258  tfrcllemsucaccv  6259  nntr2  6407  unsnfi  6815  addnidpig  7168  frecfzennn  10230  hashinfom  10556  hashennn  10558  hashp1i  10588  ennnfonelemg  11952  ctinfom  11977  nninfalllemn  13377