Theorem ordirr 4591

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 4586. If in the definition of ordinals df-iord 4414, we also required that membership be well-founded on any ordinal (see df-frind 4380), then we could prove ordirr 4591 without ax-setind 4586. (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 4590	. 2 |- -. A e. A
2	1	a1i 9	1 |- ( Ord A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2176   Ord word 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-sn 3639

This theorem is referenced by:  onirri  4592  nordeq  4593  ordn2lp  4594  orddisj  4595  onprc  4601  nlimsucg  4615  tfr1onlemsucfn  6428  tfr1onlemsucaccv  6429  tfrcllemsucfn  6441  tfrcllemsucaccv  6442  nntr2  6591  unsnfi  7018  nnnninfeq  7232  nninfisol  7237  addnidpig  7451  frecfzennn  10573  hashinfom  10925  hashennn  10927  hashp1i  10957  ennnfonelemg  12807  ctinfom  12832