Theorem ordirr 4457

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 4452. If in the definition of ordinals df-iord 4288, we also required that membership be well-founded on any ordinal (see df-frind 4254), then we could prove ordirr 4457 without ax-setind 4452. (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 4456	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1480   Ord word 4284

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-sn 3533

This theorem is referenced by:  onirri  4458  nordeq  4459  ordn2lp  4460  orddisj  4461  onprc  4467  nlimsucg  4481  tfr1onlemsucfn  6237  tfr1onlemsucaccv  6238  tfrcllemsucfn  6250  tfrcllemsucaccv  6251  nntr2  6399  unsnfi  6807  addnidpig  7144  frecfzennn  10199  hashinfom  10524  hashennn  10526  hashp1i  10556  ennnfonelemg  11916  ctinfom  11941  nninfalllemn  13202