Theorem ordirr 4575

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 4570. If in the definition of ordinals df-iord 4398, we also required that membership be well-founded on any ordinal (see df-frind 4364), then we could prove ordirr 4575 without ax-setind 4570. (Contributed by NM, 2-Jan-1994.)

Assertion

Ref	Expression
ordirr	|- ( Ord A -> -. A e. A )

Proof of Theorem ordirr

Step	Hyp	Ref	Expression
1		elirr 4574	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2164   Ord word 4394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-v 2762  df-dif 3156  df-sn 3625

This theorem is referenced by:  onirri  4576  nordeq  4577  ordn2lp  4578  orddisj  4579  onprc  4585  nlimsucg  4599  tfr1onlemsucfn  6395  tfr1onlemsucaccv  6396  tfrcllemsucfn  6408  tfrcllemsucaccv  6409  nntr2  6558  unsnfi  6977  nnnninfeq  7189  nninfisol  7194  addnidpig  7398  frecfzennn  10500  hashinfom  10852  hashennn  10854  hashp1i  10884  ennnfonelemg  12563  ctinfom  12588