Theorem ordirr 4590

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 4585. If in the definition of ordinals df-iord 4413, we also required that membership be well-founded on any ordinal (see df-frind 4379), then we could prove ordirr 4590 without ax-setind 4585. (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 4589	. 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 4409

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 4585

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  4591  nordeq  4592  ordn2lp  4593  orddisj  4594  onprc  4600  nlimsucg  4614  tfr1onlemsucfn  6426  tfr1onlemsucaccv  6427  tfrcllemsucfn  6439  tfrcllemsucaccv  6440  nntr2  6589  unsnfi  7016  nnnninfeq  7230  nninfisol  7235  addnidpig  7449  frecfzennn  10571  hashinfom  10923  hashennn  10925  hashp1i  10955  ennnfonelemg  12774  ctinfom  12799