Theorem ordirr 4499

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 4494. If in the definition of ordinals df-iord 4325, we also required that membership be well-founded on any ordinal (see df-frind 4291), then we could prove ordirr 4499 without ax-setind 4494. (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 4498	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2128   Ord word 4321

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4494

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-sn 3566

This theorem is referenced by:  onirri  4500  nordeq  4501  ordn2lp  4502  orddisj  4503  onprc  4509  nlimsucg  4523  tfr1onlemsucfn  6281  tfr1onlemsucaccv  6282  tfrcllemsucfn  6294  tfrcllemsucaccv  6295  nntr2  6443  unsnfi  6856  addnidpig  7239  frecfzennn  10307  hashinfom  10634  hashennn  10636  hashp1i  10666  ennnfonelemg  12104  ctinfom  12129  nninfalllemn  13541